Qualified Homomorphisms, Normal Kernels, Fibers and Injectivity
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-foundations · 2026-10-15 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.
Field-order and source-binding data
1. Qualify Both Groups and Every Product Pair before Applying Kernel Theorems
Use complete source Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], source identity=0, target table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], target identity=0, and total mapping=[0, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).
View answer and explanation
(3, 3, [0, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 2, 2, Yes], [0, 2, 1, 1, Yes], [1, 0, 2, 2, Yes], [1, 1, 1, 1, Yes], [1, 2, 0, 0, Yes], [2, 0, 1, 1, Yes], [2, 1, 0, 0, Yes], [2, 2, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)
List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (3, 3, [0, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 2, 2, Yes], [0, 2, 1, 1, Yes], [1, 0, 2, 2, Yes], [1, 1, 1, 1, Yes], [1, 2, 0, 0, Yes], [2, 0, 1, 1, Yes], [2, 1, 0, 0, Yes], [2, 2, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-homomorphism-qualification · candidate
2. Qualify Both Groups and Every Product Pair before Applying Kernel Theorems
Use complete source Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], source identity=0, target table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], target identity=0, and total mapping=[0, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).
View answer and explanation
(4, 4, [0, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 3, 3, Yes], [0, 2, 2, 2, Yes], [0, 3, 1, 1, Yes], [1, 0, 3, 3, Yes], [1, 1, 2, 2, Yes], [1, 2, 1, 1, Yes], [1, 3, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 1, 1, Yes], [2, 2, 0, 0, Yes], [2, 3, 3, 3, Yes], [3, 0, 1, 1, Yes], [3, 1, 0, 0, Yes], [3, 2, 3, 3, Yes], [3, 3, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)
List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (4, 4, [0, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 3, 3, Yes], [0, 2, 2, 2, Yes], [0, 3, 1, 1, Yes], [1, 0, 3, 3, Yes], [1, 1, 2, 2, Yes], [1, 2, 1, 1, Yes], [1, 3, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 1, 1, Yes], [2, 2, 0, 0, Yes], [2, 3, 3, 3, Yes], [3, 0, 1, 1, Yes], [3, 1, 0, 0, Yes], [3, 2, 3, 3, Yes], [3, 3, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-homomorphism-qualification · candidate
3. Qualify Both Groups and Every Product Pair before Applying Kernel Theorems
Use complete source Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], source identity=0, target table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], target identity=0, and total mapping=[0, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).
View answer and explanation
(5, 5, [0, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 4, 4, Yes], [0, 2, 3, 3, Yes], [0, 3, 2, 2, Yes], [0, 4, 1, 1, Yes], [1, 0, 4, 4, Yes], [1, 1, 3, 3, Yes], [1, 2, 2, 2, Yes], [1, 3, 1, 1, Yes], [1, 4, 0, 0, Yes], [2, 0, 3, 3, Yes], [2, 1, 2, 2, Yes], [2, 2, 1, 1, Yes], [2, 3, 0, 0, Yes], [2, 4, 4, 4, Yes], [3, 0, 2, 2, Yes], [3, 1, 1, 1, Yes], [3, 2, 0, 0, Yes], [3, 3, 4, 4, Yes], [3, 4, 3, 3, Yes], [4, 0, 1, 1, Yes], [4, 1, 0, 0, Yes], [4, 2, 4, 4, Yes], [4, 3, 3, 3, Yes], [4, 4, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)
List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (5, 5, [0, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 4, 4, Yes], [0, 2, 3, 3, Yes], [0, 3, 2, 2, Yes], [0, 4, 1, 1, Yes], [1, 0, 4, 4, Yes], [1, 1, 3, 3, Yes], [1, 2, 2, 2, Yes], [1, 3, 1, 1, Yes], [1, 4, 0, 0, Yes], [2, 0, 3, 3, Yes], [2, 1, 2, 2, Yes], [2, 2, 1, 1, Yes], [2, 3, 0, 0, Yes], [2, 4, 4, 4, Yes], [3, 0, 2, 2, Yes], [3, 1, 1, 1, Yes], [3, 2, 0, 0, Yes], [3, 3, 4, 4, Yes], [3, 4, 3, 3, Yes], [4, 0, 1, 1, Yes], [4, 1, 0, 0, Yes], [4, 2, 4, 4, Yes], [4, 3, 3, 3, Yes], [4, 4, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-homomorphism-qualification · candidate
4. Qualify Both Groups and Every Product Pair before Applying Kernel Theorems
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], source identity=0, target table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], target identity=0, and total mapping=[0, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).
View answer and explanation
(6, 6, [0, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 5, 5, Yes], [0, 2, 4, 4, Yes], [0, 3, 3, 3, Yes], [0, 4, 2, 2, Yes], [0, 5, 1, 1, Yes], [1, 0, 5, 5, Yes], [1, 1, 4, 4, Yes], [1, 2, 3, 3, Yes], [1, 3, 2, 2, Yes], [1, 4, 1, 1, Yes], [1, 5, 0, 0, Yes], [2, 0, 4, 4, Yes], [2, 1, 3, 3, Yes], [2, 2, 2, 2, Yes], [2, 3, 1, 1, Yes], [2, 4, 0, 0, Yes], [2, 5, 5, 5, Yes], [3, 0, 3, 3, Yes], [3, 1, 2, 2, Yes], [3, 2, 1, 1, Yes], [3, 3, 0, 0, Yes], [3, 4, 5, 5, Yes], [3, 5, 4, 4, Yes], [4, 0, 2, 2, Yes], [4, 1, 1, 1, Yes], [4, 2, 0, 0, Yes], [4, 3, 5, 5, Yes], [4, 4, 4, 4, Yes], [4, 5, 3, 3, Yes], [5, 0, 1, 1, Yes], [5, 1, 0, 0, Yes], [5, 2, 5, 5, Yes], [5, 3, 4, 4, Yes], [5, 4, 3, 3, Yes], [5, 5, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)
List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (6, 6, [0, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 5, 5, Yes], [0, 2, 4, 4, Yes], [0, 3, 3, 3, Yes], [0, 4, 2, 2, Yes], [0, 5, 1, 1, Yes], [1, 0, 5, 5, Yes], [1, 1, 4, 4, Yes], [1, 2, 3, 3, Yes], [1, 3, 2, 2, Yes], [1, 4, 1, 1, Yes], [1, 5, 0, 0, Yes], [2, 0, 4, 4, Yes], [2, 1, 3, 3, Yes], [2, 2, 2, 2, Yes], [2, 3, 1, 1, Yes], [2, 4, 0, 0, Yes], [2, 5, 5, 5, Yes], [3, 0, 3, 3, Yes], [3, 1, 2, 2, Yes], [3, 2, 1, 1, Yes], [3, 3, 0, 0, Yes], [3, 4, 5, 5, Yes], [3, 5, 4, 4, Yes], [4, 0, 2, 2, Yes], [4, 1, 1, 1, Yes], [4, 2, 0, 0, Yes], [4, 3, 5, 5, Yes], [4, 4, 4, 4, Yes], [4, 5, 3, 3, Yes], [5, 0, 1, 1, Yes], [5, 1, 0, 0, Yes], [5, 2, 5, 5, Yes], [5, 3, 4, 4, Yes], [5, 4, 3, 3, Yes], [5, 5, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-homomorphism-qualification · candidate
5. Qualify Both Groups and Every Product Pair before Applying Kernel Theorems
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], target identity=0, and total mapping=[0, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).
View answer and explanation
(7, 7, [0, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 6, 6, Yes], [0, 2, 5, 5, Yes], [0, 3, 4, 4, Yes], [0, 4, 3, 3, Yes], [0, 5, 2, 2, Yes], [0, 6, 1, 1, Yes], [1, 0, 6, 6, Yes], [1, 1, 5, 5, Yes], [1, 2, 4, 4, Yes], [1, 3, 3, 3, Yes], [1, 4, 2, 2, Yes], [1, 5, 1, 1, Yes], [1, 6, 0, 0, Yes], [2, 0, 5, 5, Yes], [2, 1, 4, 4, Yes], [2, 2, 3, 3, Yes], [2, 3, 2, 2, Yes], [2, 4, 1, 1, Yes], [2, 5, 0, 0, Yes], [2, 6, 6, 6, Yes], [3, 0, 4, 4, Yes], [3, 1, 3, 3, Yes], [3, 2, 2, 2, Yes], [3, 3, 1, 1, Yes], [3, 4, 0, 0, Yes], [3, 5, 6, 6, Yes], [3, 6, 5, 5, Yes], [4, 0, 3, 3, Yes], [4, 1, 2, 2, Yes], [4, 2, 1, 1, Yes], [4, 3, 0, 0, Yes], [4, 4, 6, 6, Yes], [4, 5, 5, 5, Yes], [4, 6, 4, 4, Yes], [5, 0, 2, 2, Yes], [5, 1, 1, 1, Yes], [5, 2, 0, 0, Yes], [5, 3, 6, 6, Yes], [5, 4, 5, 5, Yes], [5, 5, 4, 4, Yes], [5, 6, 3, 3, Yes], [6, 0, 1, 1, Yes], [6, 1, 0, 0, Yes], [6, 2, 6, 6, Yes], [6, 3, 5, 5, Yes], [6, 4, 4, 4, Yes], [6, 5, 3, 3, Yes], [6, 6, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)
List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (7, 7, [0, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 6, 6, Yes], [0, 2, 5, 5, Yes], [0, 3, 4, 4, Yes], [0, 4, 3, 3, Yes], [0, 5, 2, 2, Yes], [0, 6, 1, 1, Yes], [1, 0, 6, 6, Yes], [1, 1, 5, 5, Yes], [1, 2, 4, 4, Yes], [1, 3, 3, 3, Yes], [1, 4, 2, 2, Yes], [1, 5, 1, 1, Yes], [1, 6, 0, 0, Yes], [2, 0, 5, 5, Yes], [2, 1, 4, 4, Yes], [2, 2, 3, 3, Yes], [2, 3, 2, 2, Yes], [2, 4, 1, 1, Yes], [2, 5, 0, 0, Yes], [2, 6, 6, 6, Yes], [3, 0, 4, 4, Yes], [3, 1, 3, 3, Yes], [3, 2, 2, 2, Yes], [3, 3, 1, 1, Yes], [3, 4, 0, 0, Yes], [3, 5, 6, 6, Yes], [3, 6, 5, 5, Yes], [4, 0, 3, 3, Yes], [4, 1, 2, 2, Yes], [4, 2, 1, 1, Yes], [4, 3, 0, 0, Yes], [4, 4, 6, 6, Yes], [4, 5, 5, 5, Yes], [4, 6, 4, 4, Yes], [5, 0, 2, 2, Yes], [5, 1, 1, 1, Yes], [5, 2, 0, 0, Yes], [5, 3, 6, 6, Yes], [5, 4, 5, 5, Yes], [5, 5, 4, 4, Yes], [5, 6, 3, 3, Yes], [6, 0, 1, 1, Yes], [6, 1, 0, 0, Yes], [6, 2, 6, 6, Yes], [6, 3, 5, 5, Yes], [6, 4, 4, 4, Yes], [6, 5, 3, 3, Yes], [6, 6, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-homomorphism-qualification · candidate
6. Qualify Both Groups and Every Product Pair before Applying Kernel Theorems
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3], [5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3], [5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6]], target identity=0, and total mapping=[0, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).
View answer and explanation
(8, 8, [0, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 7, 7, Yes], [0, 2, 6, 6, Yes], [0, 3, 5, 5, Yes], [0, 4, 4, 4, Yes], [0, 5, 3, 3, Yes], [0, 6, 2, 2, Yes], [0, 7, 1, 1, Yes], [1, 0, 7, 7, Yes], [1, 1, 6, 6, Yes], [1, 2, 5, 5, Yes], [1, 3, 4, 4, Yes], [1, 4, 3, 3, Yes], [1, 5, 2, 2, Yes], [1, 6, 1, 1, Yes], [1, 7, 0, 0, Yes], [2, 0, 6, 6, Yes], [2, 1, 5, 5, Yes], [2, 2, 4, 4, Yes], [2, 3, 3, 3, Yes], [2, 4, 2, 2, Yes], [2, 5, 1, 1, Yes], [2, 6, 0, 0, Yes], [2, 7, 7, 7, Yes], [3, 0, 5, 5, Yes], [3, 1, 4, 4, Yes], [3, 2, 3, 3, Yes], [3, 3, 2, 2, Yes], [3, 4, 1, 1, Yes], [3, 5, 0, 0, Yes], [3, 6, 7, 7, Yes], [3, 7, 6, 6, Yes], [4, 0, 4, 4, Yes], [4, 1, 3, 3, Yes], [4, 2, 2, 2, Yes], [4, 3, 1, 1, Yes], [4, 4, 0, 0, Yes], [4, 5, 7, 7, Yes], [4, 6, 6, 6, Yes], [4, 7, 5, 5, Yes], [5, 0, 3, 3, Yes], [5, 1, 2, 2, Yes], [5, 2, 1, 1, Yes], [5, 3, 0, 0, Yes], [5, 4, 7, 7, Yes], [5, 5, 6, 6, Yes], [5, 6, 5, 5, Yes], [5, 7, 4, 4, Yes], [6, 0, 2, 2, Yes], [6, 1, 1, 1, Yes], [6, 2, 0, 0, Yes], [6, 3, 7, 7, Yes], [6, 4, 6, 6, Yes], [6, 5, 5, 5, Yes], [6, 6, 4, 4, Yes], [6, 7, 3, 3, Yes], [7, 0, 1, 1, Yes], [7, 1, 0, 0, Yes], [7, 2, 7, 7, Yes], [7, 3, 6, 6, Yes], [7, 4, 5, 5, Yes], [7, 5, 4, 4, Yes], [7, 6, 3, 3, Yes], [7, 7, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)
List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (8, 8, [0, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 7, 7, Yes], [0, 2, 6, 6, Yes], [0, 3, 5, 5, Yes], [0, 4, 4, 4, Yes], [0, 5, 3, 3, Yes], [0, 6, 2, 2, Yes], [0, 7, 1, 1, Yes], [1, 0, 7, 7, Yes], [1, 1, 6, 6, Yes], [1, 2, 5, 5, Yes], [1, 3, 4, 4, Yes], [1, 4, 3, 3, Yes], [1, 5, 2, 2, Yes], [1, 6, 1, 1, Yes], [1, 7, 0, 0, Yes], [2, 0, 6, 6, Yes], [2, 1, 5, 5, Yes], [2, 2, 4, 4, Yes], [2, 3, 3, 3, Yes], [2, 4, 2, 2, Yes], [2, 5, 1, 1, Yes], [2, 6, 0, 0, Yes], [2, 7, 7, 7, Yes], [3, 0, 5, 5, Yes], [3, 1, 4, 4, Yes], [3, 2, 3, 3, Yes], [3, 3, 2, 2, Yes], [3, 4, 1, 1, Yes], [3, 5, 0, 0, Yes], [3, 6, 7, 7, Yes], [3, 7, 6, 6, Yes], [4, 0, 4, 4, Yes], [4, 1, 3, 3, Yes], [4, 2, 2, 2, Yes], [4, 3, 1, 1, Yes], [4, 4, 0, 0, Yes], [4, 5, 7, 7, Yes], [4, 6, 6, 6, Yes], [4, 7, 5, 5, Yes], [5, 0, 3, 3, Yes], [5, 1, 2, 2, Yes], [5, 2, 1, 1, Yes], [5, 3, 0, 0, Yes], [5, 4, 7, 7, Yes], [5, 5, 6, 6, Yes], [5, 6, 5, 5, Yes], [5, 7, 4, 4, Yes], [6, 0, 2, 2, Yes], [6, 1, 1, 1, Yes], [6, 2, 0, 0, Yes], [6, 3, 7, 7, Yes], [6, 4, 6, 6, Yes], [6, 5, 5, 5, Yes], [6, 6, 4, 4, Yes], [6, 7, 3, 3, Yes], [7, 0, 1, 1, Yes], [7, 1, 0, 0, Yes], [7, 2, 7, 7, Yes], [7, 3, 6, 6, Yes], [7, 4, 5, 5, Yes], [7, 5, 4, 4, Yes], [7, 6, 3, 3, Yes], [7, 7, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-homomorphism-qualification · candidate
7. Qualify Both Groups and Every Product Pair before Applying Kernel Theorems
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 6, 7, 8, 0], [2, 3, 4, 5, 6, 7, 8, 0, 1], [3, 4, 5, 6, 7, 8, 0, 1, 2], [4, 5, 6, 7, 8, 0, 1, 2, 3], [5, 6, 7, 8, 0, 1, 2, 3, 4], [6, 7, 8, 0, 1, 2, 3, 4, 5], [7, 8, 0, 1, 2, 3, 4, 5, 6], [8, 0, 1, 2, 3, 4, 5, 6, 7]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 6, 7, 8, 0], [2, 3, 4, 5, 6, 7, 8, 0, 1], [3, 4, 5, 6, 7, 8, 0, 1, 2], [4, 5, 6, 7, 8, 0, 1, 2, 3], [5, 6, 7, 8, 0, 1, 2, 3, 4], [6, 7, 8, 0, 1, 2, 3, 4, 5], [7, 8, 0, 1, 2, 3, 4, 5, 6], [8, 0, 1, 2, 3, 4, 5, 6, 7]], target identity=0, and total mapping=[0, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).
View answer and explanation
(9, 9, [0, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 8, 8, Yes], [0, 2, 7, 7, Yes], [0, 3, 6, 6, Yes], [0, 4, 5, 5, Yes], [0, 5, 4, 4, Yes], [0, 6, 3, 3, Yes], [0, 7, 2, 2, Yes], [0, 8, 1, 1, Yes], [1, 0, 8, 8, Yes], [1, 1, 7, 7, Yes], [1, 2, 6, 6, Yes], [1, 3, 5, 5, Yes], [1, 4, 4, 4, Yes], [1, 5, 3, 3, Yes], [1, 6, 2, 2, Yes], [1, 7, 1, 1, Yes], [1, 8, 0, 0, Yes], [2, 0, 7, 7, Yes], [2, 1, 6, 6, Yes], [2, 2, 5, 5, Yes], [2, 3, 4, 4, Yes], [2, 4, 3, 3, Yes], [2, 5, 2, 2, Yes], [2, 6, 1, 1, Yes], [2, 7, 0, 0, Yes], [2, 8, 8, 8, Yes], [3, 0, 6, 6, Yes], [3, 1, 5, 5, Yes], [3, 2, 4, 4, Yes], [3, 3, 3, 3, Yes], [3, 4, 2, 2, Yes], [3, 5, 1, 1, Yes], [3, 6, 0, 0, Yes], [3, 7, 8, 8, Yes], [3, 8, 7, 7, Yes], [4, 0, 5, 5, Yes], [4, 1, 4, 4, Yes], [4, 2, 3, 3, Yes], [4, 3, 2, 2, Yes], [4, 4, 1, 1, Yes], [4, 5, 0, 0, Yes], [4, 6, 8, 8, Yes], [4, 7, 7, 7, Yes], [4, 8, 6, 6, Yes], [5, 0, 4, 4, Yes], [5, 1, 3, 3, Yes], [5, 2, 2, 2, Yes], [5, 3, 1, 1, Yes], [5, 4, 0, 0, Yes], [5, 5, 8, 8, Yes], [5, 6, 7, 7, Yes], [5, 7, 6, 6, Yes], [5, 8, 5, 5, Yes], [6, 0, 3, 3, Yes], [6, 1, 2, 2, Yes], [6, 2, 1, 1, Yes], [6, 3, 0, 0, Yes], [6, 4, 8, 8, Yes], [6, 5, 7, 7, Yes], [6, 6, 6, 6, Yes], [6, 7, 5, 5, Yes], [6, 8, 4, 4, Yes], [7, 0, 2, 2, Yes], [7, 1, 1, 1, Yes], [7, 2, 0, 0, Yes], [7, 3, 8, 8, Yes], [7, 4, 7, 7, Yes], [7, 5, 6, 6, Yes], [7, 6, 5, 5, Yes], [7, 7, 4, 4, Yes], [7, 8, 3, 3, Yes], [8, 0, 1, 1, Yes], [8, 1, 0, 0, Yes], [8, 2, 8, 8, Yes], [8, 3, 7, 7, Yes], [8, 4, 6, 6, Yes], [8, 5, 5, 5, Yes], [8, 6, 4, 4, Yes], [8, 7, 3, 3, Yes], [8, 8, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)
List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (9, 9, [0, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 8, 8, Yes], [0, 2, 7, 7, Yes], [0, 3, 6, 6, Yes], [0, 4, 5, 5, Yes], [0, 5, 4, 4, Yes], [0, 6, 3, 3, Yes], [0, 7, 2, 2, Yes], [0, 8, 1, 1, Yes], [1, 0, 8, 8, Yes], [1, 1, 7, 7, Yes], [1, 2, 6, 6, Yes], [1, 3, 5, 5, Yes], [1, 4, 4, 4, Yes], [1, 5, 3, 3, Yes], [1, 6, 2, 2, Yes], [1, 7, 1, 1, Yes], [1, 8, 0, 0, Yes], [2, 0, 7, 7, Yes], [2, 1, 6, 6, Yes], [2, 2, 5, 5, Yes], [2, 3, 4, 4, Yes], [2, 4, 3, 3, Yes], [2, 5, 2, 2, Yes], [2, 6, 1, 1, Yes], [2, 7, 0, 0, Yes], [2, 8, 8, 8, Yes], [3, 0, 6, 6, Yes], [3, 1, 5, 5, Yes], [3, 2, 4, 4, Yes], [3, 3, 3, 3, Yes], [3, 4, 2, 2, Yes], [3, 5, 1, 1, Yes], [3, 6, 0, 0, Yes], [3, 7, 8, 8, Yes], [3, 8, 7, 7, Yes], [4, 0, 5, 5, Yes], [4, 1, 4, 4, Yes], [4, 2, 3, 3, Yes], [4, 3, 2, 2, Yes], [4, 4, 1, 1, Yes], [4, 5, 0, 0, Yes], [4, 6, 8, 8, Yes], [4, 7, 7, 7, Yes], [4, 8, 6, 6, Yes], [5, 0, 4, 4, Yes], [5, 1, 3, 3, Yes], [5, 2, 2, 2, Yes], [5, 3, 1, 1, Yes], [5, 4, 0, 0, Yes], [5, 5, 8, 8, Yes], [5, 6, 7, 7, Yes], [5, 7, 6, 6, Yes], [5, 8, 5, 5, Yes], [6, 0, 3, 3, Yes], [6, 1, 2, 2, Yes], [6, 2, 1, 1, Yes], [6, 3, 0, 0, Yes], [6, 4, 8, 8, Yes], [6, 5, 7, 7, Yes], [6, 6, 6, 6, Yes], [6, 7, 5, 5, Yes], [6, 8, 4, 4, Yes], [7, 0, 2, 2, Yes], [7, 1, 1, 1, Yes], [7, 2, 0, 0, Yes], [7, 3, 8, 8, Yes], [7, 4, 7, 7, Yes], [7, 5, 6, 6, Yes], [7, 6, 5, 5, Yes], [7, 7, 4, 4, Yes], [7, 8, 3, 3, Yes], [8, 0, 1, 1, Yes], [8, 1, 0, 0, Yes], [8, 2, 8, 8, Yes], [8, 3, 7, 7, Yes], [8, 4, 6, 6, Yes], [8, 5, 5, 5, Yes], [8, 6, 4, 4, Yes], [8, 7, 3, 3, Yes], [8, 8, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-homomorphism-qualification · candidate
8. Qualify Both Groups and Every Product Pair before Applying Kernel Theorems
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8]], target identity=0, and total mapping=[0, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).
View answer and explanation
(10, 10, [0, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 9, 9, Yes], [0, 2, 8, 8, Yes], [0, 3, 7, 7, Yes], [0, 4, 6, 6, Yes], [0, 5, 5, 5, Yes], [0, 6, 4, 4, Yes], [0, 7, 3, 3, Yes], [0, 8, 2, 2, Yes], [0, 9, 1, 1, Yes], [1, 0, 9, 9, Yes], [1, 1, 8, 8, Yes], [1, 2, 7, 7, Yes], [1, 3, 6, 6, Yes], [1, 4, 5, 5, Yes], [1, 5, 4, 4, Yes], [1, 6, 3, 3, Yes], [1, 7, 2, 2, Yes], [1, 8, 1, 1, Yes], [1, 9, 0, 0, Yes], [2, 0, 8, 8, Yes], [2, 1, 7, 7, Yes], [2, 2, 6, 6, Yes], [2, 3, 5, 5, Yes], [2, 4, 4, 4, Yes], [2, 5, 3, 3, Yes], [2, 6, 2, 2, Yes], [2, 7, 1, 1, Yes], [2, 8, 0, 0, Yes], [2, 9, 9, 9, Yes], [3, 0, 7, 7, Yes], [3, 1, 6, 6, Yes], [3, 2, 5, 5, Yes], [3, 3, 4, 4, Yes], [3, 4, 3, 3, Yes], [3, 5, 2, 2, Yes], [3, 6, 1, 1, Yes], [3, 7, 0, 0, Yes], [3, 8, 9, 9, Yes], [3, 9, 8, 8, Yes], [4, 0, 6, 6, Yes], [4, 1, 5, 5, Yes], [4, 2, 4, 4, Yes], [4, 3, 3, 3, Yes], [4, 4, 2, 2, Yes], [4, 5, 1, 1, Yes], [4, 6, 0, 0, Yes], [4, 7, 9, 9, Yes], [4, 8, 8, 8, Yes], [4, 9, 7, 7, Yes], [5, 0, 5, 5, Yes], [5, 1, 4, 4, Yes], [5, 2, 3, 3, Yes], [5, 3, 2, 2, Yes], [5, 4, 1, 1, Yes], [5, 5, 0, 0, Yes], [5, 6, 9, 9, Yes], [5, 7, 8, 8, Yes], [5, 8, 7, 7, Yes], [5, 9, 6, 6, Yes], [6, 0, 4, 4, Yes], [6, 1, 3, 3, Yes], [6, 2, 2, 2, Yes], [6, 3, 1, 1, Yes], [6, 4, 0, 0, Yes], [6, 5, 9, 9, Yes], [6, 6, 8, 8, Yes], [6, 7, 7, 7, Yes], [6, 8, 6, 6, Yes], [6, 9, 5, 5, Yes], [7, 0, 3, 3, Yes], [7, 1, 2, 2, Yes], [7, 2, 1, 1, Yes], [7, 3, 0, 0, Yes], [7, 4, 9, 9, Yes], [7, 5, 8, 8, Yes], [7, 6, 7, 7, Yes], [7, 7, 6, 6, Yes], [7, 8, 5, 5, Yes], [7, 9, 4, 4, Yes], [8, 0, 2, 2, Yes], [8, 1, 1, 1, Yes], [8, 2, 0, 0, Yes], [8, 3, 9, 9, Yes], [8, 4, 8, 8, Yes], [8, 5, 7, 7, Yes], [8, 6, 6, 6, Yes], [8, 7, 5, 5, Yes], [8, 8, 4, 4, Yes], [8, 9, 3, 3, Yes], [9, 0, 1, 1, Yes], [9, 1, 0, 0, Yes], [9, 2, 9, 9, Yes], [9, 3, 8, 8, Yes], [9, 4, 7, 7, Yes], [9, 5, 6, 6, Yes], [9, 6, 5, 5, Yes], [9, 7, 4, 4, Yes], [9, 8, 3, 3, Yes], [9, 9, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)
List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (10, 10, [0, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 9, 9, Yes], [0, 2, 8, 8, Yes], [0, 3, 7, 7, Yes], [0, 4, 6, 6, Yes], [0, 5, 5, 5, Yes], [0, 6, 4, 4, Yes], [0, 7, 3, 3, Yes], [0, 8, 2, 2, Yes], [0, 9, 1, 1, Yes], [1, 0, 9, 9, Yes], [1, 1, 8, 8, Yes], [1, 2, 7, 7, Yes], [1, 3, 6, 6, Yes], [1, 4, 5, 5, Yes], [1, 5, 4, 4, Yes], [1, 6, 3, 3, Yes], [1, 7, 2, 2, Yes], [1, 8, 1, 1, Yes], [1, 9, 0, 0, Yes], [2, 0, 8, 8, Yes], [2, 1, 7, 7, Yes], [2, 2, 6, 6, Yes], [2, 3, 5, 5, Yes], [2, 4, 4, 4, Yes], [2, 5, 3, 3, Yes], [2, 6, 2, 2, Yes], [2, 7, 1, 1, Yes], [2, 8, 0, 0, Yes], [2, 9, 9, 9, Yes], [3, 0, 7, 7, Yes], [3, 1, 6, 6, Yes], [3, 2, 5, 5, Yes], [3, 3, 4, 4, Yes], [3, 4, 3, 3, Yes], [3, 5, 2, 2, Yes], [3, 6, 1, 1, Yes], [3, 7, 0, 0, Yes], [3, 8, 9, 9, Yes], [3, 9, 8, 8, Yes], [4, 0, 6, 6, Yes], [4, 1, 5, 5, Yes], [4, 2, 4, 4, Yes], [4, 3, 3, 3, Yes], [4, 4, 2, 2, Yes], [4, 5, 1, 1, Yes], [4, 6, 0, 0, Yes], [4, 7, 9, 9, Yes], [4, 8, 8, 8, Yes], [4, 9, 7, 7, Yes], [5, 0, 5, 5, Yes], [5, 1, 4, 4, Yes], [5, 2, 3, 3, Yes], [5, 3, 2, 2, Yes], [5, 4, 1, 1, Yes], [5, 5, 0, 0, Yes], [5, 6, 9, 9, Yes], [5, 7, 8, 8, Yes], [5, 8, 7, 7, Yes], [5, 9, 6, 6, Yes], [6, 0, 4, 4, Yes], [6, 1, 3, 3, Yes], [6, 2, 2, 2, Yes], [6, 3, 1, 1, Yes], [6, 4, 0, 0, Yes], [6, 5, 9, 9, Yes], [6, 6, 8, 8, Yes], [6, 7, 7, 7, Yes], [6, 8, 6, 6, Yes], [6, 9, 5, 5, Yes], [7, 0, 3, 3, Yes], [7, 1, 2, 2, Yes], [7, 2, 1, 1, Yes], [7, 3, 0, 0, Yes], [7, 4, 9, 9, Yes], [7, 5, 8, 8, Yes], [7, 6, 7, 7, Yes], [7, 7, 6, 6, Yes], [7, 8, 5, 5, Yes], [7, 9, 4, 4, Yes], [8, 0, 2, 2, Yes], [8, 1, 1, 1, Yes], [8, 2, 0, 0, Yes], [8, 3, 9, 9, Yes], [8, 4, 8, 8, Yes], [8, 5, 7, 7, Yes], [8, 6, 6, 6, Yes], [8, 7, 5, 5, Yes], [8, 8, 4, 4, Yes], [8, 9, 3, 3, Yes], [9, 0, 1, 1, Yes], [9, 1, 0, 0, Yes], [9, 2, 9, 9, Yes], [9, 3, 8, 8, Yes], [9, 4, 7, 7, Yes], [9, 5, 6, 6, Yes], [9, 6, 5, 5, Yes], [9, 7, 4, 4, Yes], [9, 8, 3, 3, Yes], [9, 9, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-homomorphism-qualification · candidate
9. Qualify Both Groups and Every Product Pair before Applying Kernel Theorems
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9]], target identity=0, and total mapping=[0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).
View answer and explanation
(11, 11, [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 10, 10, Yes], [0, 2, 9, 9, Yes], [0, 3, 8, 8, Yes], [0, 4, 7, 7, Yes], [0, 5, 6, 6, Yes], [0, 6, 5, 5, Yes], [0, 7, 4, 4, Yes], [0, 8, 3, 3, Yes], [0, 9, 2, 2, Yes], [0, 10, 1, 1, Yes], [1, 0, 10, 10, Yes], [1, 1, 9, 9, Yes], [1, 2, 8, 8, Yes], [1, 3, 7, 7, Yes], [1, 4, 6, 6, Yes], [1, 5, 5, 5, Yes], [1, 6, 4, 4, Yes], [1, 7, 3, 3, Yes], [1, 8, 2, 2, Yes], [1, 9, 1, 1, Yes], [1, 10, 0, 0, Yes], [2, 0, 9, 9, Yes], [2, 1, 8, 8, Yes], [2, 2, 7, 7, Yes], [2, 3, 6, 6, Yes], [2, 4, 5, 5, Yes], [2, 5, 4, 4, Yes], [2, 6, 3, 3, Yes], [2, 7, 2, 2, Yes], [2, 8, 1, 1, Yes], [2, 9, 0, 0, Yes], [2, 10, 10, 10, Yes], [3, 0, 8, 8, Yes], [3, 1, 7, 7, Yes], [3, 2, 6, 6, Yes], [3, 3, 5, 5, Yes], [3, 4, 4, 4, Yes], [3, 5, 3, 3, Yes], [3, 6, 2, 2, Yes], [3, 7, 1, 1, Yes], [3, 8, 0, 0, Yes], [3, 9, 10, 10, Yes], [3, 10, 9, 9, Yes], [4, 0, 7, 7, Yes], [4, 1, 6, 6, Yes], [4, 2, 5, 5, Yes], [4, 3, 4, 4, Yes], [4, 4, 3, 3, Yes], [4, 5, 2, 2, Yes], [4, 6, 1, 1, Yes], [4, 7, 0, 0, Yes], [4, 8, 10, 10, Yes], [4, 9, 9, 9, Yes], [4, 10, 8, 8, Yes], [5, 0, 6, 6, Yes], [5, 1, 5, 5, Yes], [5, 2, 4, 4, Yes], [5, 3, 3, 3, Yes], [5, 4, 2, 2, Yes], [5, 5, 1, 1, Yes], [5, 6, 0, 0, Yes], [5, 7, 10, 10, Yes], [5, 8, 9, 9, Yes], [5, 9, 8, 8, Yes], [5, 10, 7, 7, Yes], [6, 0, 5, 5, Yes], [6, 1, 4, 4, Yes], [6, 2, 3, 3, Yes], [6, 3, 2, 2, Yes], [6, 4, 1, 1, Yes], [6, 5, 0, 0, Yes], [6, 6, 10, 10, Yes], [6, 7, 9, 9, Yes], [6, 8, 8, 8, Yes], [6, 9, 7, 7, Yes], [6, 10, 6, 6, Yes], [7, 0, 4, 4, Yes], [7, 1, 3, 3, Yes], [7, 2, 2, 2, Yes], [7, 3, 1, 1, Yes], [7, 4, 0, 0, Yes], [7, 5, 10, 10, Yes], [7, 6, 9, 9, Yes], [7, 7, 8, 8, Yes], [7, 8, 7, 7, Yes], [7, 9, 6, 6, Yes], [7, 10, 5, 5, Yes], [8, 0, 3, 3, Yes], [8, 1, 2, 2, Yes], [8, 2, 1, 1, Yes], [8, 3, 0, 0, Yes], [8, 4, 10, 10, Yes], [8, 5, 9, 9, Yes], [8, 6, 8, 8, Yes], [8, 7, 7, 7, Yes], [8, 8, 6, 6, Yes], [8, 9, 5, 5, Yes], [8, 10, 4, 4, Yes], [9, 0, 2, 2, Yes], [9, 1, 1, 1, Yes], [9, 2, 0, 0, Yes], [9, 3, 10, 10, Yes], [9, 4, 9, 9, Yes], [9, 5, 8, 8, Yes], [9, 6, 7, 7, Yes], [9, 7, 6, 6, Yes], [9, 8, 5, 5, Yes], [9, 9, 4, 4, Yes], [9, 10, 3, 3, Yes], [10, 0, 1, 1, Yes], [10, 1, 0, 0, Yes], [10, 2, 10, 10, Yes], [10, 3, 9, 9, Yes], [10, 4, 8, 8, Yes], [10, 5, 7, 7, Yes], [10, 6, 6, 6, Yes], [10, 7, 5, 5, Yes], [10, 8, 4, 4, Yes], [10, 9, 3, 3, Yes], [10, 10, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)
List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (11, 11, [0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 10, 10, Yes], [0, 2, 9, 9, Yes], [0, 3, 8, 8, Yes], [0, 4, 7, 7, Yes], [0, 5, 6, 6, Yes], [0, 6, 5, 5, Yes], [0, 7, 4, 4, Yes], [0, 8, 3, 3, Yes], [0, 9, 2, 2, Yes], [0, 10, 1, 1, Yes], [1, 0, 10, 10, Yes], [1, 1, 9, 9, Yes], [1, 2, 8, 8, Yes], [1, 3, 7, 7, Yes], [1, 4, 6, 6, Yes], [1, 5, 5, 5, Yes], [1, 6, 4, 4, Yes], [1, 7, 3, 3, Yes], [1, 8, 2, 2, Yes], [1, 9, 1, 1, Yes], [1, 10, 0, 0, Yes], [2, 0, 9, 9, Yes], [2, 1, 8, 8, Yes], [2, 2, 7, 7, Yes], [2, 3, 6, 6, Yes], [2, 4, 5, 5, Yes], [2, 5, 4, 4, Yes], [2, 6, 3, 3, Yes], [2, 7, 2, 2, Yes], [2, 8, 1, 1, Yes], [2, 9, 0, 0, Yes], [2, 10, 10, 10, Yes], [3, 0, 8, 8, Yes], [3, 1, 7, 7, Yes], [3, 2, 6, 6, Yes], [3, 3, 5, 5, Yes], [3, 4, 4, 4, Yes], [3, 5, 3, 3, Yes], [3, 6, 2, 2, Yes], [3, 7, 1, 1, Yes], [3, 8, 0, 0, Yes], [3, 9, 10, 10, Yes], [3, 10, 9, 9, Yes], [4, 0, 7, 7, Yes], [4, 1, 6, 6, Yes], [4, 2, 5, 5, Yes], [4, 3, 4, 4, Yes], [4, 4, 3, 3, Yes], [4, 5, 2, 2, Yes], [4, 6, 1, 1, Yes], [4, 7, 0, 0, Yes], [4, 8, 10, 10, Yes], [4, 9, 9, 9, Yes], [4, 10, 8, 8, Yes], [5, 0, 6, 6, Yes], [5, 1, 5, 5, Yes], [5, 2, 4, 4, Yes], [5, 3, 3, 3, Yes], [5, 4, 2, 2, Yes], [5, 5, 1, 1, Yes], [5, 6, 0, 0, Yes], [5, 7, 10, 10, Yes], [5, 8, 9, 9, Yes], [5, 9, 8, 8, Yes], [5, 10, 7, 7, Yes], [6, 0, 5, 5, Yes], [6, 1, 4, 4, Yes], [6, 2, 3, 3, Yes], [6, 3, 2, 2, Yes], [6, 4, 1, 1, Yes], [6, 5, 0, 0, Yes], [6, 6, 10, 10, Yes], [6, 7, 9, 9, Yes], [6, 8, 8, 8, Yes], [6, 9, 7, 7, Yes], [6, 10, 6, 6, Yes], [7, 0, 4, 4, Yes], [7, 1, 3, 3, Yes], [7, 2, 2, 2, Yes], [7, 3, 1, 1, Yes], [7, 4, 0, 0, Yes], [7, 5, 10, 10, Yes], [7, 6, 9, 9, Yes], [7, 7, 8, 8, Yes], [7, 8, 7, 7, Yes], [7, 9, 6, 6, Yes], [7, 10, 5, 5, Yes], [8, 0, 3, 3, Yes], [8, 1, 2, 2, Yes], [8, 2, 1, 1, Yes], [8, 3, 0, 0, Yes], [8, 4, 10, 10, Yes], [8, 5, 9, 9, Yes], [8, 6, 8, 8, Yes], [8, 7, 7, 7, Yes], [8, 8, 6, 6, Yes], [8, 9, 5, 5, Yes], [8, 10, 4, 4, Yes], [9, 0, 2, 2, Yes], [9, 1, 1, 1, Yes], [9, 2, 0, 0, Yes], [9, 3, 10, 10, Yes], [9, 4, 9, 9, Yes], [9, 5, 8, 8, Yes], [9, 6, 7, 7, Yes], [9, 7, 6, 6, Yes], [9, 8, 5, 5, Yes], [9, 9, 4, 4, Yes], [9, 10, 3, 3, Yes], [10, 0, 1, 1, Yes], [10, 1, 0, 0, Yes], [10, 2, 10, 10, Yes], [10, 3, 9, 9, Yes], [10, 4, 8, 8, Yes], [10, 5, 7, 7, Yes], [10, 6, 6, 6, Yes], [10, 7, 5, 5, Yes], [10, 8, 4, 4, Yes], [10, 9, 3, 3, Yes], [10, 10, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-homomorphism-qualification · candidate
10. Qualify Both Groups and Every Product Pair before Applying Kernel Theorems
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]], target identity=0, and total mapping=[0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).
View answer and explanation
(12, 12, [0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 11, 11, Yes], [0, 2, 10, 10, Yes], [0, 3, 9, 9, Yes], [0, 4, 8, 8, Yes], [0, 5, 7, 7, Yes], [0, 6, 6, 6, Yes], [0, 7, 5, 5, Yes], [0, 8, 4, 4, Yes], [0, 9, 3, 3, Yes], [0, 10, 2, 2, Yes], [0, 11, 1, 1, Yes], [1, 0, 11, 11, Yes], [1, 1, 10, 10, Yes], [1, 2, 9, 9, Yes], [1, 3, 8, 8, Yes], [1, 4, 7, 7, Yes], [1, 5, 6, 6, Yes], [1, 6, 5, 5, Yes], [1, 7, 4, 4, Yes], [1, 8, 3, 3, Yes], [1, 9, 2, 2, Yes], [1, 10, 1, 1, Yes], [1, 11, 0, 0, Yes], [2, 0, 10, 10, Yes], [2, 1, 9, 9, Yes], [2, 2, 8, 8, Yes], [2, 3, 7, 7, Yes], [2, 4, 6, 6, Yes], [2, 5, 5, 5, Yes], [2, 6, 4, 4, Yes], [2, 7, 3, 3, Yes], [2, 8, 2, 2, Yes], [2, 9, 1, 1, Yes], [2, 10, 0, 0, Yes], [2, 11, 11, 11, Yes], [3, 0, 9, 9, Yes], [3, 1, 8, 8, Yes], [3, 2, 7, 7, Yes], [3, 3, 6, 6, Yes], [3, 4, 5, 5, Yes], [3, 5, 4, 4, Yes], [3, 6, 3, 3, Yes], [3, 7, 2, 2, Yes], [3, 8, 1, 1, Yes], [3, 9, 0, 0, Yes], [3, 10, 11, 11, Yes], [3, 11, 10, 10, Yes], [4, 0, 8, 8, Yes], [4, 1, 7, 7, Yes], [4, 2, 6, 6, Yes], [4, 3, 5, 5, Yes], [4, 4, 4, 4, Yes], [4, 5, 3, 3, Yes], [4, 6, 2, 2, Yes], [4, 7, 1, 1, Yes], [4, 8, 0, 0, Yes], [4, 9, 11, 11, Yes], [4, 10, 10, 10, Yes], [4, 11, 9, 9, Yes], [5, 0, 7, 7, Yes], [5, 1, 6, 6, Yes], [5, 2, 5, 5, Yes], [5, 3, 4, 4, Yes], [5, 4, 3, 3, Yes], [5, 5, 2, 2, Yes], [5, 6, 1, 1, Yes], [5, 7, 0, 0, Yes], [5, 8, 11, 11, Yes], [5, 9, 10, 10, Yes], [5, 10, 9, 9, Yes], [5, 11, 8, 8, Yes], [6, 0, 6, 6, Yes], [6, 1, 5, 5, Yes], [6, 2, 4, 4, Yes], [6, 3, 3, 3, Yes], [6, 4, 2, 2, Yes], [6, 5, 1, 1, Yes], [6, 6, 0, 0, Yes], [6, 7, 11, 11, Yes], [6, 8, 10, 10, Yes], [6, 9, 9, 9, Yes], [6, 10, 8, 8, Yes], [6, 11, 7, 7, Yes], [7, 0, 5, 5, Yes], [7, 1, 4, 4, Yes], [7, 2, 3, 3, Yes], [7, 3, 2, 2, Yes], [7, 4, 1, 1, Yes], [7, 5, 0, 0, Yes], [7, 6, 11, 11, Yes], [7, 7, 10, 10, Yes], [7, 8, 9, 9, Yes], [7, 9, 8, 8, Yes], [7, 10, 7, 7, Yes], [7, 11, 6, 6, Yes], [8, 0, 4, 4, Yes], [8, 1, 3, 3, Yes], [8, 2, 2, 2, Yes], [8, 3, 1, 1, Yes], [8, 4, 0, 0, Yes], [8, 5, 11, 11, Yes], [8, 6, 10, 10, Yes], [8, 7, 9, 9, Yes], [8, 8, 8, 8, Yes], [8, 9, 7, 7, Yes], [8, 10, 6, 6, Yes], [8, 11, 5, 5, Yes], [9, 0, 3, 3, Yes], [9, 1, 2, 2, Yes], [9, 2, 1, 1, Yes], [9, 3, 0, 0, Yes], [9, 4, 11, 11, Yes], [9, 5, 10, 10, Yes], [9, 6, 9, 9, Yes], [9, 7, 8, 8, Yes], [9, 8, 7, 7, Yes], [9, 9, 6, 6, Yes], [9, 10, 5, 5, Yes], [9, 11, 4, 4, Yes], [10, 0, 2, 2, Yes], [10, 1, 1, 1, Yes], [10, 2, 0, 0, Yes], [10, 3, 11, 11, Yes], [10, 4, 10, 10, Yes], [10, 5, 9, 9, Yes], [10, 6, 8, 8, Yes], [10, 7, 7, 7, Yes], [10, 8, 6, 6, Yes], [10, 9, 5, 5, Yes], [10, 10, 4, 4, Yes], [10, 11, 3, 3, Yes], [11, 0, 1, 1, Yes], [11, 1, 0, 0, Yes], [11, 2, 11, 11, Yes], [11, 3, 10, 10, Yes], [11, 4, 9, 9, Yes], [11, 5, 8, 8, Yes], [11, 6, 7, 7, Yes], [11, 7, 6, 6, Yes], [11, 8, 5, 5, Yes], [11, 9, 4, 4, Yes], [11, 10, 3, 3, Yes], [11, 11, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)
List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (12, 12, [0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 11, 11, Yes], [0, 2, 10, 10, Yes], [0, 3, 9, 9, Yes], [0, 4, 8, 8, Yes], [0, 5, 7, 7, Yes], [0, 6, 6, 6, Yes], [0, 7, 5, 5, Yes], [0, 8, 4, 4, Yes], [0, 9, 3, 3, Yes], [0, 10, 2, 2, Yes], [0, 11, 1, 1, Yes], [1, 0, 11, 11, Yes], [1, 1, 10, 10, Yes], [1, 2, 9, 9, Yes], [1, 3, 8, 8, Yes], [1, 4, 7, 7, Yes], [1, 5, 6, 6, Yes], [1, 6, 5, 5, Yes], [1, 7, 4, 4, Yes], [1, 8, 3, 3, Yes], [1, 9, 2, 2, Yes], [1, 10, 1, 1, Yes], [1, 11, 0, 0, Yes], [2, 0, 10, 10, Yes], [2, 1, 9, 9, Yes], [2, 2, 8, 8, Yes], [2, 3, 7, 7, Yes], [2, 4, 6, 6, Yes], [2, 5, 5, 5, Yes], [2, 6, 4, 4, Yes], [2, 7, 3, 3, Yes], [2, 8, 2, 2, Yes], [2, 9, 1, 1, Yes], [2, 10, 0, 0, Yes], [2, 11, 11, 11, Yes], [3, 0, 9, 9, Yes], [3, 1, 8, 8, Yes], [3, 2, 7, 7, Yes], [3, 3, 6, 6, Yes], [3, 4, 5, 5, Yes], [3, 5, 4, 4, Yes], [3, 6, 3, 3, Yes], [3, 7, 2, 2, Yes], [3, 8, 1, 1, Yes], [3, 9, 0, 0, Yes], [3, 10, 11, 11, Yes], [3, 11, 10, 10, Yes], [4, 0, 8, 8, Yes], [4, 1, 7, 7, Yes], [4, 2, 6, 6, Yes], [4, 3, 5, 5, Yes], [4, 4, 4, 4, Yes], [4, 5, 3, 3, Yes], [4, 6, 2, 2, Yes], [4, 7, 1, 1, Yes], [4, 8, 0, 0, Yes], [4, 9, 11, 11, Yes], [4, 10, 10, 10, Yes], [4, 11, 9, 9, Yes], [5, 0, 7, 7, Yes], [5, 1, 6, 6, Yes], [5, 2, 5, 5, Yes], [5, 3, 4, 4, Yes], [5, 4, 3, 3, Yes], [5, 5, 2, 2, Yes], [5, 6, 1, 1, Yes], [5, 7, 0, 0, Yes], [5, 8, 11, 11, Yes], [5, 9, 10, 10, Yes], [5, 10, 9, 9, Yes], [5, 11, 8, 8, Yes], [6, 0, 6, 6, Yes], [6, 1, 5, 5, Yes], [6, 2, 4, 4, Yes], [6, 3, 3, 3, Yes], [6, 4, 2, 2, Yes], [6, 5, 1, 1, Yes], [6, 6, 0, 0, Yes], [6, 7, 11, 11, Yes], [6, 8, 10, 10, Yes], [6, 9, 9, 9, Yes], [6, 10, 8, 8, Yes], [6, 11, 7, 7, Yes], [7, 0, 5, 5, Yes], [7, 1, 4, 4, Yes], [7, 2, 3, 3, Yes], [7, 3, 2, 2, Yes], [7, 4, 1, 1, Yes], [7, 5, 0, 0, Yes], [7, 6, 11, 11, Yes], [7, 7, 10, 10, Yes], [7, 8, 9, 9, Yes], [7, 9, 8, 8, Yes], [7, 10, 7, 7, Yes], [7, 11, 6, 6, Yes], [8, 0, 4, 4, Yes], [8, 1, 3, 3, Yes], [8, 2, 2, 2, Yes], [8, 3, 1, 1, Yes], [8, 4, 0, 0, Yes], [8, 5, 11, 11, Yes], [8, 6, 10, 10, Yes], [8, 7, 9, 9, Yes], [8, 8, 8, 8, Yes], [8, 9, 7, 7, Yes], [8, 10, 6, 6, Yes], [8, 11, 5, 5, Yes], [9, 0, 3, 3, Yes], [9, 1, 2, 2, Yes], [9, 2, 1, 1, Yes], [9, 3, 0, 0, Yes], [9, 4, 11, 11, Yes], [9, 5, 10, 10, Yes], [9, 6, 9, 9, Yes], [9, 7, 8, 8, Yes], [9, 8, 7, 7, Yes], [9, 9, 6, 6, Yes], [9, 10, 5, 5, Yes], [9, 11, 4, 4, Yes], [10, 0, 2, 2, Yes], [10, 1, 1, 1, Yes], [10, 2, 0, 0, Yes], [10, 3, 11, 11, Yes], [10, 4, 10, 10, Yes], [10, 5, 9, 9, Yes], [10, 6, 8, 8, Yes], [10, 7, 7, 7, Yes], [10, 8, 6, 6, Yes], [10, 9, 5, 5, Yes], [10, 10, 4, 4, Yes], [10, 11, 3, 3, Yes], [11, 0, 1, 1, Yes], [11, 1, 0, 0, Yes], [11, 2, 11, 11, Yes], [11, 3, 10, 10, Yes], [11, 4, 9, 9, Yes], [11, 5, 8, 8, Yes], [11, 6, 7, 7, Yes], [11, 7, 6, 6, Yes], [11, 8, 5, 5, Yes], [11, 9, 4, 4, Yes], [11, 10, 3, 3, Yes], [11, 11, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-homomorphism-qualification · candidate
11. Qualify Both Groups and Every Product Pair before Applying Kernel Theorems
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]], target identity=0, and total mapping=[0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).
View answer and explanation
(13, 13, [0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 12, 12, Yes], [0, 2, 11, 11, Yes], [0, 3, 10, 10, Yes], [0, 4, 9, 9, Yes], [0, 5, 8, 8, Yes], [0, 6, 7, 7, Yes], [0, 7, 6, 6, Yes], [0, 8, 5, 5, Yes], [0, 9, 4, 4, Yes], [0, 10, 3, 3, Yes], [0, 11, 2, 2, Yes], [0, 12, 1, 1, Yes], [1, 0, 12, 12, Yes], [1, 1, 11, 11, Yes], [1, 2, 10, 10, Yes], [1, 3, 9, 9, Yes], [1, 4, 8, 8, Yes], [1, 5, 7, 7, Yes], [1, 6, 6, 6, Yes], [1, 7, 5, 5, Yes], [1, 8, 4, 4, Yes], [1, 9, 3, 3, Yes], [1, 10, 2, 2, Yes], [1, 11, 1, 1, Yes], [1, 12, 0, 0, Yes], [2, 0, 11, 11, Yes], [2, 1, 10, 10, Yes], [2, 2, 9, 9, Yes], [2, 3, 8, 8, Yes], [2, 4, 7, 7, Yes], [2, 5, 6, 6, Yes], [2, 6, 5, 5, Yes], [2, 7, 4, 4, Yes], [2, 8, 3, 3, Yes], [2, 9, 2, 2, Yes], [2, 10, 1, 1, Yes], [2, 11, 0, 0, Yes], [2, 12, 12, 12, Yes], [3, 0, 10, 10, Yes], [3, 1, 9, 9, Yes], [3, 2, 8, 8, Yes], [3, 3, 7, 7, Yes], [3, 4, 6, 6, Yes], [3, 5, 5, 5, Yes], [3, 6, 4, 4, Yes], [3, 7, 3, 3, Yes], [3, 8, 2, 2, Yes], [3, 9, 1, 1, Yes], [3, 10, 0, 0, Yes], [3, 11, 12, 12, Yes], [3, 12, 11, 11, Yes], [4, 0, 9, 9, Yes], [4, 1, 8, 8, Yes], [4, 2, 7, 7, Yes], [4, 3, 6, 6, Yes], [4, 4, 5, 5, Yes], [4, 5, 4, 4, Yes], [4, 6, 3, 3, Yes], [4, 7, 2, 2, Yes], [4, 8, 1, 1, Yes], [4, 9, 0, 0, Yes], [4, 10, 12, 12, Yes], [4, 11, 11, 11, Yes], [4, 12, 10, 10, Yes], [5, 0, 8, 8, Yes], [5, 1, 7, 7, Yes], [5, 2, 6, 6, Yes], [5, 3, 5, 5, Yes], [5, 4, 4, 4, Yes], [5, 5, 3, 3, Yes], [5, 6, 2, 2, Yes], [5, 7, 1, 1, Yes], [5, 8, 0, 0, Yes], [5, 9, 12, 12, Yes], [5, 10, 11, 11, Yes], [5, 11, 10, 10, Yes], [5, 12, 9, 9, Yes], [6, 0, 7, 7, Yes], [6, 1, 6, 6, Yes], [6, 2, 5, 5, Yes], [6, 3, 4, 4, Yes], [6, 4, 3, 3, Yes], [6, 5, 2, 2, Yes], [6, 6, 1, 1, Yes], [6, 7, 0, 0, Yes], [6, 8, 12, 12, Yes], [6, 9, 11, 11, Yes], [6, 10, 10, 10, Yes], [6, 11, 9, 9, Yes], [6, 12, 8, 8, Yes], [7, 0, 6, 6, Yes], [7, 1, 5, 5, Yes], [7, 2, 4, 4, Yes], [7, 3, 3, 3, Yes], [7, 4, 2, 2, Yes], [7, 5, 1, 1, Yes], [7, 6, 0, 0, Yes], [7, 7, 12, 12, Yes], [7, 8, 11, 11, Yes], [7, 9, 10, 10, Yes], [7, 10, 9, 9, Yes], [7, 11, 8, 8, Yes], [7, 12, 7, 7, Yes], [8, 0, 5, 5, Yes], [8, 1, 4, 4, Yes], [8, 2, 3, 3, Yes], [8, 3, 2, 2, Yes], [8, 4, 1, 1, Yes], [8, 5, 0, 0, Yes], [8, 6, 12, 12, Yes], [8, 7, 11, 11, Yes], [8, 8, 10, 10, Yes], [8, 9, 9, 9, Yes], [8, 10, 8, 8, Yes], [8, 11, 7, 7, Yes], [8, 12, 6, 6, Yes], [9, 0, 4, 4, Yes], [9, 1, 3, 3, Yes], [9, 2, 2, 2, Yes], [9, 3, 1, 1, Yes], [9, 4, 0, 0, Yes], [9, 5, 12, 12, Yes], [9, 6, 11, 11, Yes], [9, 7, 10, 10, Yes], [9, 8, 9, 9, Yes], [9, 9, 8, 8, Yes], [9, 10, 7, 7, Yes], [9, 11, 6, 6, Yes], [9, 12, 5, 5, Yes], [10, 0, 3, 3, Yes], [10, 1, 2, 2, Yes], [10, 2, 1, 1, Yes], [10, 3, 0, 0, Yes], [10, 4, 12, 12, Yes], [10, 5, 11, 11, Yes], [10, 6, 10, 10, Yes], [10, 7, 9, 9, Yes], [10, 8, 8, 8, Yes], [10, 9, 7, 7, Yes], [10, 10, 6, 6, Yes], [10, 11, 5, 5, Yes], [10, 12, 4, 4, Yes], [11, 0, 2, 2, Yes], [11, 1, 1, 1, Yes], [11, 2, 0, 0, Yes], [11, 3, 12, 12, Yes], [11, 4, 11, 11, Yes], [11, 5, 10, 10, Yes], [11, 6, 9, 9, Yes], [11, 7, 8, 8, Yes], [11, 8, 7, 7, Yes], [11, 9, 6, 6, Yes], [11, 10, 5, 5, Yes], [11, 11, 4, 4, Yes], [11, 12, 3, 3, Yes], [12, 0, 1, 1, Yes], [12, 1, 0, 0, Yes], [12, 2, 12, 12, Yes], [12, 3, 11, 11, Yes], [12, 4, 10, 10, Yes], [12, 5, 9, 9, Yes], [12, 6, 8, 8, Yes], [12, 7, 7, 7, Yes], [12, 8, 6, 6, Yes], [12, 9, 5, 5, Yes], [12, 10, 4, 4, Yes], [12, 11, 3, 3, Yes], [12, 12, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)
List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (13, 13, [0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 12, 12, Yes], [0, 2, 11, 11, Yes], [0, 3, 10, 10, Yes], [0, 4, 9, 9, Yes], [0, 5, 8, 8, Yes], [0, 6, 7, 7, Yes], [0, 7, 6, 6, Yes], [0, 8, 5, 5, Yes], [0, 9, 4, 4, Yes], [0, 10, 3, 3, Yes], [0, 11, 2, 2, Yes], [0, 12, 1, 1, Yes], [1, 0, 12, 12, Yes], [1, 1, 11, 11, Yes], [1, 2, 10, 10, Yes], [1, 3, 9, 9, Yes], [1, 4, 8, 8, Yes], [1, 5, 7, 7, Yes], [1, 6, 6, 6, Yes], [1, 7, 5, 5, Yes], [1, 8, 4, 4, Yes], [1, 9, 3, 3, Yes], [1, 10, 2, 2, Yes], [1, 11, 1, 1, Yes], [1, 12, 0, 0, Yes], [2, 0, 11, 11, Yes], [2, 1, 10, 10, Yes], [2, 2, 9, 9, Yes], [2, 3, 8, 8, Yes], [2, 4, 7, 7, Yes], [2, 5, 6, 6, Yes], [2, 6, 5, 5, Yes], [2, 7, 4, 4, Yes], [2, 8, 3, 3, Yes], [2, 9, 2, 2, Yes], [2, 10, 1, 1, Yes], [2, 11, 0, 0, Yes], [2, 12, 12, 12, Yes], [3, 0, 10, 10, Yes], [3, 1, 9, 9, Yes], [3, 2, 8, 8, Yes], [3, 3, 7, 7, Yes], [3, 4, 6, 6, Yes], [3, 5, 5, 5, Yes], [3, 6, 4, 4, Yes], [3, 7, 3, 3, Yes], [3, 8, 2, 2, Yes], [3, 9, 1, 1, Yes], [3, 10, 0, 0, Yes], [3, 11, 12, 12, Yes], [3, 12, 11, 11, Yes], [4, 0, 9, 9, Yes], [4, 1, 8, 8, Yes], [4, 2, 7, 7, Yes], [4, 3, 6, 6, Yes], [4, 4, 5, 5, Yes], [4, 5, 4, 4, Yes], [4, 6, 3, 3, Yes], [4, 7, 2, 2, Yes], [4, 8, 1, 1, Yes], [4, 9, 0, 0, Yes], [4, 10, 12, 12, Yes], [4, 11, 11, 11, Yes], [4, 12, 10, 10, Yes], [5, 0, 8, 8, Yes], [5, 1, 7, 7, Yes], [5, 2, 6, 6, Yes], [5, 3, 5, 5, Yes], [5, 4, 4, 4, Yes], [5, 5, 3, 3, Yes], [5, 6, 2, 2, Yes], [5, 7, 1, 1, Yes], [5, 8, 0, 0, Yes], [5, 9, 12, 12, Yes], [5, 10, 11, 11, Yes], [5, 11, 10, 10, Yes], [5, 12, 9, 9, Yes], [6, 0, 7, 7, Yes], [6, 1, 6, 6, Yes], [6, 2, 5, 5, Yes], [6, 3, 4, 4, Yes], [6, 4, 3, 3, Yes], [6, 5, 2, 2, Yes], [6, 6, 1, 1, Yes], [6, 7, 0, 0, Yes], [6, 8, 12, 12, Yes], [6, 9, 11, 11, Yes], [6, 10, 10, 10, Yes], [6, 11, 9, 9, Yes], [6, 12, 8, 8, Yes], [7, 0, 6, 6, Yes], [7, 1, 5, 5, Yes], [7, 2, 4, 4, Yes], [7, 3, 3, 3, Yes], [7, 4, 2, 2, Yes], [7, 5, 1, 1, Yes], [7, 6, 0, 0, Yes], [7, 7, 12, 12, Yes], [7, 8, 11, 11, Yes], [7, 9, 10, 10, Yes], [7, 10, 9, 9, Yes], [7, 11, 8, 8, Yes], [7, 12, 7, 7, Yes], [8, 0, 5, 5, Yes], [8, 1, 4, 4, Yes], [8, 2, 3, 3, Yes], [8, 3, 2, 2, Yes], [8, 4, 1, 1, Yes], [8, 5, 0, 0, Yes], [8, 6, 12, 12, Yes], [8, 7, 11, 11, Yes], [8, 8, 10, 10, Yes], [8, 9, 9, 9, Yes], [8, 10, 8, 8, Yes], [8, 11, 7, 7, Yes], [8, 12, 6, 6, Yes], [9, 0, 4, 4, Yes], [9, 1, 3, 3, Yes], [9, 2, 2, 2, Yes], [9, 3, 1, 1, Yes], [9, 4, 0, 0, Yes], [9, 5, 12, 12, Yes], [9, 6, 11, 11, Yes], [9, 7, 10, 10, Yes], [9, 8, 9, 9, Yes], [9, 9, 8, 8, Yes], [9, 10, 7, 7, Yes], [9, 11, 6, 6, Yes], [9, 12, 5, 5, Yes], [10, 0, 3, 3, Yes], [10, 1, 2, 2, Yes], [10, 2, 1, 1, Yes], [10, 3, 0, 0, Yes], [10, 4, 12, 12, Yes], [10, 5, 11, 11, Yes], [10, 6, 10, 10, Yes], [10, 7, 9, 9, Yes], [10, 8, 8, 8, Yes], [10, 9, 7, 7, Yes], [10, 10, 6, 6, Yes], [10, 11, 5, 5, Yes], [10, 12, 4, 4, Yes], [11, 0, 2, 2, Yes], [11, 1, 1, 1, Yes], [11, 2, 0, 0, Yes], [11, 3, 12, 12, Yes], [11, 4, 11, 11, Yes], [11, 5, 10, 10, Yes], [11, 6, 9, 9, Yes], [11, 7, 8, 8, Yes], [11, 8, 7, 7, Yes], [11, 9, 6, 6, Yes], [11, 10, 5, 5, Yes], [11, 11, 4, 4, Yes], [11, 12, 3, 3, Yes], [12, 0, 1, 1, Yes], [12, 1, 0, 0, Yes], [12, 2, 12, 12, Yes], [12, 3, 11, 11, Yes], [12, 4, 10, 10, Yes], [12, 5, 9, 9, Yes], [12, 6, 8, 8, Yes], [12, 7, 7, 7, Yes], [12, 8, 6, 6, Yes], [12, 9, 5, 5, Yes], [12, 10, 4, 4, Yes], [12, 11, 3, 3, Yes], [12, 12, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-homomorphism-qualification · candidate
12. Qualify Both Groups and Every Product Pair before Applying Kernel Theorems
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]], target identity=0, and total mapping=[0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).
View answer and explanation
(14, 14, [0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 13, 13, Yes], [0, 2, 12, 12, Yes], [0, 3, 11, 11, Yes], [0, 4, 10, 10, Yes], [0, 5, 9, 9, Yes], [0, 6, 8, 8, Yes], [0, 7, 7, 7, Yes], [0, 8, 6, 6, Yes], [0, 9, 5, 5, Yes], [0, 10, 4, 4, Yes], [0, 11, 3, 3, Yes], [0, 12, 2, 2, Yes], [0, 13, 1, 1, Yes], [1, 0, 13, 13, Yes], [1, 1, 12, 12, Yes], [1, 2, 11, 11, Yes], [1, 3, 10, 10, Yes], [1, 4, 9, 9, Yes], [1, 5, 8, 8, Yes], [1, 6, 7, 7, Yes], [1, 7, 6, 6, Yes], [1, 8, 5, 5, Yes], [1, 9, 4, 4, Yes], [1, 10, 3, 3, Yes], [1, 11, 2, 2, Yes], [1, 12, 1, 1, Yes], [1, 13, 0, 0, Yes], [2, 0, 12, 12, Yes], [2, 1, 11, 11, Yes], [2, 2, 10, 10, Yes], [2, 3, 9, 9, Yes], [2, 4, 8, 8, Yes], [2, 5, 7, 7, Yes], [2, 6, 6, 6, Yes], [2, 7, 5, 5, Yes], [2, 8, 4, 4, Yes], [2, 9, 3, 3, Yes], [2, 10, 2, 2, Yes], [2, 11, 1, 1, Yes], [2, 12, 0, 0, Yes], [2, 13, 13, 13, Yes], [3, 0, 11, 11, Yes], [3, 1, 10, 10, Yes], [3, 2, 9, 9, Yes], [3, 3, 8, 8, Yes], [3, 4, 7, 7, Yes], [3, 5, 6, 6, Yes], [3, 6, 5, 5, Yes], [3, 7, 4, 4, Yes], [3, 8, 3, 3, Yes], [3, 9, 2, 2, Yes], [3, 10, 1, 1, Yes], [3, 11, 0, 0, Yes], [3, 12, 13, 13, Yes], [3, 13, 12, 12, Yes], [4, 0, 10, 10, Yes], [4, 1, 9, 9, Yes], [4, 2, 8, 8, Yes], [4, 3, 7, 7, Yes], [4, 4, 6, 6, Yes], [4, 5, 5, 5, Yes], [4, 6, 4, 4, Yes], [4, 7, 3, 3, Yes], [4, 8, 2, 2, Yes], [4, 9, 1, 1, Yes], [4, 10, 0, 0, Yes], [4, 11, 13, 13, Yes], [4, 12, 12, 12, Yes], [4, 13, 11, 11, Yes], [5, 0, 9, 9, Yes], [5, 1, 8, 8, Yes], [5, 2, 7, 7, Yes], [5, 3, 6, 6, Yes], [5, 4, 5, 5, Yes], [5, 5, 4, 4, Yes], [5, 6, 3, 3, Yes], [5, 7, 2, 2, Yes], [5, 8, 1, 1, Yes], [5, 9, 0, 0, Yes], [5, 10, 13, 13, Yes], [5, 11, 12, 12, Yes], [5, 12, 11, 11, Yes], [5, 13, 10, 10, Yes], [6, 0, 8, 8, Yes], [6, 1, 7, 7, Yes], [6, 2, 6, 6, Yes], [6, 3, 5, 5, Yes], [6, 4, 4, 4, Yes], [6, 5, 3, 3, Yes], [6, 6, 2, 2, Yes], [6, 7, 1, 1, Yes], [6, 8, 0, 0, Yes], [6, 9, 13, 13, Yes], [6, 10, 12, 12, Yes], [6, 11, 11, 11, Yes], [6, 12, 10, 10, Yes], [6, 13, 9, 9, Yes], [7, 0, 7, 7, Yes], [7, 1, 6, 6, Yes], [7, 2, 5, 5, Yes], [7, 3, 4, 4, Yes], [7, 4, 3, 3, Yes], [7, 5, 2, 2, Yes], [7, 6, 1, 1, Yes], [7, 7, 0, 0, Yes], [7, 8, 13, 13, Yes], [7, 9, 12, 12, Yes], [7, 10, 11, 11, Yes], [7, 11, 10, 10, Yes], [7, 12, 9, 9, Yes], [7, 13, 8, 8, Yes], [8, 0, 6, 6, Yes], [8, 1, 5, 5, Yes], [8, 2, 4, 4, Yes], [8, 3, 3, 3, Yes], [8, 4, 2, 2, Yes], [8, 5, 1, 1, Yes], [8, 6, 0, 0, Yes], [8, 7, 13, 13, Yes], [8, 8, 12, 12, Yes], [8, 9, 11, 11, Yes], [8, 10, 10, 10, Yes], [8, 11, 9, 9, Yes], [8, 12, 8, 8, Yes], [8, 13, 7, 7, Yes], [9, 0, 5, 5, Yes], [9, 1, 4, 4, Yes], [9, 2, 3, 3, Yes], [9, 3, 2, 2, Yes], [9, 4, 1, 1, Yes], [9, 5, 0, 0, Yes], [9, 6, 13, 13, Yes], [9, 7, 12, 12, Yes], [9, 8, 11, 11, Yes], [9, 9, 10, 10, Yes], [9, 10, 9, 9, Yes], [9, 11, 8, 8, Yes], [9, 12, 7, 7, Yes], [9, 13, 6, 6, Yes], [10, 0, 4, 4, Yes], [10, 1, 3, 3, Yes], [10, 2, 2, 2, Yes], [10, 3, 1, 1, Yes], [10, 4, 0, 0, Yes], [10, 5, 13, 13, Yes], [10, 6, 12, 12, Yes], [10, 7, 11, 11, Yes], [10, 8, 10, 10, Yes], [10, 9, 9, 9, Yes], [10, 10, 8, 8, Yes], [10, 11, 7, 7, Yes], [10, 12, 6, 6, Yes], [10, 13, 5, 5, Yes], [11, 0, 3, 3, Yes], [11, 1, 2, 2, Yes], [11, 2, 1, 1, Yes], [11, 3, 0, 0, Yes], [11, 4, 13, 13, Yes], [11, 5, 12, 12, Yes], [11, 6, 11, 11, Yes], [11, 7, 10, 10, Yes], [11, 8, 9, 9, Yes], [11, 9, 8, 8, Yes], [11, 10, 7, 7, Yes], [11, 11, 6, 6, Yes], [11, 12, 5, 5, Yes], [11, 13, 4, 4, Yes], [12, 0, 2, 2, Yes], [12, 1, 1, 1, Yes], [12, 2, 0, 0, Yes], [12, 3, 13, 13, Yes], [12, 4, 12, 12, Yes], [12, 5, 11, 11, Yes], [12, 6, 10, 10, Yes], [12, 7, 9, 9, Yes], [12, 8, 8, 8, Yes], [12, 9, 7, 7, Yes], [12, 10, 6, 6, Yes], [12, 11, 5, 5, Yes], [12, 12, 4, 4, Yes], [12, 13, 3, 3, Yes], [13, 0, 1, 1, Yes], [13, 1, 0, 0, Yes], [13, 2, 13, 13, Yes], [13, 3, 12, 12, Yes], [13, 4, 11, 11, Yes], [13, 5, 10, 10, Yes], [13, 6, 9, 9, Yes], [13, 7, 8, 8, Yes], [13, 8, 7, 7, Yes], [13, 9, 6, 6, Yes], [13, 10, 5, 5, Yes], [13, 11, 4, 4, Yes], [13, 12, 3, 3, Yes], [13, 13, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)
List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (14, 14, [0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 13, 13, Yes], [0, 2, 12, 12, Yes], [0, 3, 11, 11, Yes], [0, 4, 10, 10, Yes], [0, 5, 9, 9, Yes], [0, 6, 8, 8, Yes], [0, 7, 7, 7, Yes], [0, 8, 6, 6, Yes], [0, 9, 5, 5, Yes], [0, 10, 4, 4, Yes], [0, 11, 3, 3, Yes], [0, 12, 2, 2, Yes], [0, 13, 1, 1, Yes], [1, 0, 13, 13, Yes], [1, 1, 12, 12, Yes], [1, 2, 11, 11, Yes], [1, 3, 10, 10, Yes], [1, 4, 9, 9, Yes], [1, 5, 8, 8, Yes], [1, 6, 7, 7, Yes], [1, 7, 6, 6, Yes], [1, 8, 5, 5, Yes], [1, 9, 4, 4, Yes], [1, 10, 3, 3, Yes], [1, 11, 2, 2, Yes], [1, 12, 1, 1, Yes], [1, 13, 0, 0, Yes], [2, 0, 12, 12, Yes], [2, 1, 11, 11, Yes], [2, 2, 10, 10, Yes], [2, 3, 9, 9, Yes], [2, 4, 8, 8, Yes], [2, 5, 7, 7, Yes], [2, 6, 6, 6, Yes], [2, 7, 5, 5, Yes], [2, 8, 4, 4, Yes], [2, 9, 3, 3, Yes], [2, 10, 2, 2, Yes], [2, 11, 1, 1, Yes], [2, 12, 0, 0, Yes], [2, 13, 13, 13, Yes], [3, 0, 11, 11, Yes], [3, 1, 10, 10, Yes], [3, 2, 9, 9, Yes], [3, 3, 8, 8, Yes], [3, 4, 7, 7, Yes], [3, 5, 6, 6, Yes], [3, 6, 5, 5, Yes], [3, 7, 4, 4, Yes], [3, 8, 3, 3, Yes], [3, 9, 2, 2, Yes], [3, 10, 1, 1, Yes], [3, 11, 0, 0, Yes], [3, 12, 13, 13, Yes], [3, 13, 12, 12, Yes], [4, 0, 10, 10, Yes], [4, 1, 9, 9, Yes], [4, 2, 8, 8, Yes], [4, 3, 7, 7, Yes], [4, 4, 6, 6, Yes], [4, 5, 5, 5, Yes], [4, 6, 4, 4, Yes], [4, 7, 3, 3, Yes], [4, 8, 2, 2, Yes], [4, 9, 1, 1, Yes], [4, 10, 0, 0, Yes], [4, 11, 13, 13, Yes], [4, 12, 12, 12, Yes], [4, 13, 11, 11, Yes], [5, 0, 9, 9, Yes], [5, 1, 8, 8, Yes], [5, 2, 7, 7, Yes], [5, 3, 6, 6, Yes], [5, 4, 5, 5, Yes], [5, 5, 4, 4, Yes], [5, 6, 3, 3, Yes], [5, 7, 2, 2, Yes], [5, 8, 1, 1, Yes], [5, 9, 0, 0, Yes], [5, 10, 13, 13, Yes], [5, 11, 12, 12, Yes], [5, 12, 11, 11, Yes], [5, 13, 10, 10, Yes], [6, 0, 8, 8, Yes], [6, 1, 7, 7, Yes], [6, 2, 6, 6, Yes], [6, 3, 5, 5, Yes], [6, 4, 4, 4, Yes], [6, 5, 3, 3, Yes], [6, 6, 2, 2, Yes], [6, 7, 1, 1, Yes], [6, 8, 0, 0, Yes], [6, 9, 13, 13, Yes], [6, 10, 12, 12, Yes], [6, 11, 11, 11, Yes], [6, 12, 10, 10, Yes], [6, 13, 9, 9, Yes], [7, 0, 7, 7, Yes], [7, 1, 6, 6, Yes], [7, 2, 5, 5, Yes], [7, 3, 4, 4, Yes], [7, 4, 3, 3, Yes], [7, 5, 2, 2, Yes], [7, 6, 1, 1, Yes], [7, 7, 0, 0, Yes], [7, 8, 13, 13, Yes], [7, 9, 12, 12, Yes], [7, 10, 11, 11, Yes], [7, 11, 10, 10, Yes], [7, 12, 9, 9, Yes], [7, 13, 8, 8, Yes], [8, 0, 6, 6, Yes], [8, 1, 5, 5, Yes], [8, 2, 4, 4, Yes], [8, 3, 3, 3, Yes], [8, 4, 2, 2, Yes], [8, 5, 1, 1, Yes], [8, 6, 0, 0, Yes], [8, 7, 13, 13, Yes], [8, 8, 12, 12, Yes], [8, 9, 11, 11, Yes], [8, 10, 10, 10, Yes], [8, 11, 9, 9, Yes], [8, 12, 8, 8, Yes], [8, 13, 7, 7, Yes], [9, 0, 5, 5, Yes], [9, 1, 4, 4, Yes], [9, 2, 3, 3, Yes], [9, 3, 2, 2, Yes], [9, 4, 1, 1, Yes], [9, 5, 0, 0, Yes], [9, 6, 13, 13, Yes], [9, 7, 12, 12, Yes], [9, 8, 11, 11, Yes], [9, 9, 10, 10, Yes], [9, 10, 9, 9, Yes], [9, 11, 8, 8, Yes], [9, 12, 7, 7, Yes], [9, 13, 6, 6, Yes], [10, 0, 4, 4, Yes], [10, 1, 3, 3, Yes], [10, 2, 2, 2, Yes], [10, 3, 1, 1, Yes], [10, 4, 0, 0, Yes], [10, 5, 13, 13, Yes], [10, 6, 12, 12, Yes], [10, 7, 11, 11, Yes], [10, 8, 10, 10, Yes], [10, 9, 9, 9, Yes], [10, 10, 8, 8, Yes], [10, 11, 7, 7, Yes], [10, 12, 6, 6, Yes], [10, 13, 5, 5, Yes], [11, 0, 3, 3, Yes], [11, 1, 2, 2, Yes], [11, 2, 1, 1, Yes], [11, 3, 0, 0, Yes], [11, 4, 13, 13, Yes], [11, 5, 12, 12, Yes], [11, 6, 11, 11, Yes], [11, 7, 10, 10, Yes], [11, 8, 9, 9, Yes], [11, 9, 8, 8, Yes], [11, 10, 7, 7, Yes], [11, 11, 6, 6, Yes], [11, 12, 5, 5, Yes], [11, 13, 4, 4, Yes], [12, 0, 2, 2, Yes], [12, 1, 1, 1, Yes], [12, 2, 0, 0, Yes], [12, 3, 13, 13, Yes], [12, 4, 12, 12, Yes], [12, 5, 11, 11, Yes], [12, 6, 10, 10, Yes], [12, 7, 9, 9, Yes], [12, 8, 8, 8, Yes], [12, 9, 7, 7, Yes], [12, 10, 6, 6, Yes], [12, 11, 5, 5, Yes], [12, 12, 4, 4, Yes], [12, 13, 3, 3, Yes], [13, 0, 1, 1, Yes], [13, 1, 0, 0, Yes], [13, 2, 13, 13, Yes], [13, 3, 12, 12, Yes], [13, 4, 11, 11, Yes], [13, 5, 10, 10, Yes], [13, 6, 9, 9, Yes], [13, 7, 8, 8, Yes], [13, 8, 7, 7, Yes], [13, 9, 6, 6, Yes], [13, 10, 5, 5, Yes], [13, 11, 4, 4, Yes], [13, 12, 3, 3, Yes], [13, 13, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-homomorphism-qualification · candidate
13. Qualify Both Groups and Every Product Pair before Applying Kernel Theorems
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]], target identity=0, and total mapping=[0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).
View answer and explanation
(15, 15, [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 14, 14, Yes], [0, 2, 13, 13, Yes], [0, 3, 12, 12, Yes], [0, 4, 11, 11, Yes], [0, 5, 10, 10, Yes], [0, 6, 9, 9, Yes], [0, 7, 8, 8, Yes], [0, 8, 7, 7, Yes], [0, 9, 6, 6, Yes], [0, 10, 5, 5, Yes], [0, 11, 4, 4, Yes], [0, 12, 3, 3, Yes], [0, 13, 2, 2, Yes], [0, 14, 1, 1, Yes], [1, 0, 14, 14, Yes], [1, 1, 13, 13, Yes], [1, 2, 12, 12, Yes], [1, 3, 11, 11, Yes], [1, 4, 10, 10, Yes], [1, 5, 9, 9, Yes], [1, 6, 8, 8, Yes], [1, 7, 7, 7, Yes], [1, 8, 6, 6, Yes], [1, 9, 5, 5, Yes], [1, 10, 4, 4, Yes], [1, 11, 3, 3, Yes], [1, 12, 2, 2, Yes], [1, 13, 1, 1, Yes], [1, 14, 0, 0, Yes], [2, 0, 13, 13, Yes], [2, 1, 12, 12, Yes], [2, 2, 11, 11, Yes], [2, 3, 10, 10, Yes], [2, 4, 9, 9, Yes], [2, 5, 8, 8, Yes], [2, 6, 7, 7, Yes], [2, 7, 6, 6, Yes], [2, 8, 5, 5, Yes], [2, 9, 4, 4, Yes], [2, 10, 3, 3, Yes], [2, 11, 2, 2, Yes], [2, 12, 1, 1, Yes], [2, 13, 0, 0, Yes], [2, 14, 14, 14, Yes], [3, 0, 12, 12, Yes], [3, 1, 11, 11, Yes], [3, 2, 10, 10, Yes], [3, 3, 9, 9, Yes], [3, 4, 8, 8, Yes], [3, 5, 7, 7, Yes], [3, 6, 6, 6, Yes], [3, 7, 5, 5, Yes], [3, 8, 4, 4, Yes], [3, 9, 3, 3, Yes], [3, 10, 2, 2, Yes], [3, 11, 1, 1, Yes], [3, 12, 0, 0, Yes], [3, 13, 14, 14, Yes], [3, 14, 13, 13, Yes], [4, 0, 11, 11, Yes], [4, 1, 10, 10, Yes], [4, 2, 9, 9, Yes], [4, 3, 8, 8, Yes], [4, 4, 7, 7, Yes], [4, 5, 6, 6, Yes], [4, 6, 5, 5, Yes], [4, 7, 4, 4, Yes], [4, 8, 3, 3, Yes], [4, 9, 2, 2, Yes], [4, 10, 1, 1, Yes], [4, 11, 0, 0, Yes], [4, 12, 14, 14, Yes], [4, 13, 13, 13, Yes], [4, 14, 12, 12, Yes], [5, 0, 10, 10, Yes], [5, 1, 9, 9, Yes], [5, 2, 8, 8, Yes], [5, 3, 7, 7, Yes], [5, 4, 6, 6, Yes], [5, 5, 5, 5, Yes], [5, 6, 4, 4, Yes], [5, 7, 3, 3, Yes], [5, 8, 2, 2, Yes], [5, 9, 1, 1, Yes], [5, 10, 0, 0, Yes], [5, 11, 14, 14, Yes], [5, 12, 13, 13, Yes], [5, 13, 12, 12, Yes], [5, 14, 11, 11, Yes], [6, 0, 9, 9, Yes], [6, 1, 8, 8, Yes], [6, 2, 7, 7, Yes], [6, 3, 6, 6, Yes], [6, 4, 5, 5, Yes], [6, 5, 4, 4, Yes], [6, 6, 3, 3, Yes], [6, 7, 2, 2, Yes], [6, 8, 1, 1, Yes], [6, 9, 0, 0, Yes], [6, 10, 14, 14, Yes], [6, 11, 13, 13, Yes], [6, 12, 12, 12, Yes], [6, 13, 11, 11, Yes], [6, 14, 10, 10, Yes], [7, 0, 8, 8, Yes], [7, 1, 7, 7, Yes], [7, 2, 6, 6, Yes], [7, 3, 5, 5, Yes], [7, 4, 4, 4, Yes], [7, 5, 3, 3, Yes], [7, 6, 2, 2, Yes], [7, 7, 1, 1, Yes], [7, 8, 0, 0, Yes], [7, 9, 14, 14, Yes], [7, 10, 13, 13, Yes], [7, 11, 12, 12, Yes], [7, 12, 11, 11, Yes], [7, 13, 10, 10, Yes], [7, 14, 9, 9, Yes], [8, 0, 7, 7, Yes], [8, 1, 6, 6, Yes], [8, 2, 5, 5, Yes], [8, 3, 4, 4, Yes], [8, 4, 3, 3, Yes], [8, 5, 2, 2, Yes], [8, 6, 1, 1, Yes], [8, 7, 0, 0, Yes], [8, 8, 14, 14, Yes], [8, 9, 13, 13, Yes], [8, 10, 12, 12, Yes], [8, 11, 11, 11, Yes], [8, 12, 10, 10, Yes], [8, 13, 9, 9, Yes], [8, 14, 8, 8, Yes], [9, 0, 6, 6, Yes], [9, 1, 5, 5, Yes], [9, 2, 4, 4, Yes], [9, 3, 3, 3, Yes], [9, 4, 2, 2, Yes], [9, 5, 1, 1, Yes], [9, 6, 0, 0, Yes], [9, 7, 14, 14, Yes], [9, 8, 13, 13, Yes], [9, 9, 12, 12, Yes], [9, 10, 11, 11, Yes], [9, 11, 10, 10, Yes], [9, 12, 9, 9, Yes], [9, 13, 8, 8, Yes], [9, 14, 7, 7, Yes], [10, 0, 5, 5, Yes], [10, 1, 4, 4, Yes], [10, 2, 3, 3, Yes], [10, 3, 2, 2, Yes], [10, 4, 1, 1, Yes], [10, 5, 0, 0, Yes], [10, 6, 14, 14, Yes], [10, 7, 13, 13, Yes], [10, 8, 12, 12, Yes], [10, 9, 11, 11, Yes], [10, 10, 10, 10, Yes], [10, 11, 9, 9, Yes], [10, 12, 8, 8, Yes], [10, 13, 7, 7, Yes], [10, 14, 6, 6, Yes], [11, 0, 4, 4, Yes], [11, 1, 3, 3, Yes], [11, 2, 2, 2, Yes], [11, 3, 1, 1, Yes], [11, 4, 0, 0, Yes], [11, 5, 14, 14, Yes], [11, 6, 13, 13, Yes], [11, 7, 12, 12, Yes], [11, 8, 11, 11, Yes], [11, 9, 10, 10, Yes], [11, 10, 9, 9, Yes], [11, 11, 8, 8, Yes], [11, 12, 7, 7, Yes], [11, 13, 6, 6, Yes], [11, 14, 5, 5, Yes], [12, 0, 3, 3, Yes], [12, 1, 2, 2, Yes], [12, 2, 1, 1, Yes], [12, 3, 0, 0, Yes], [12, 4, 14, 14, Yes], [12, 5, 13, 13, Yes], [12, 6, 12, 12, Yes], [12, 7, 11, 11, Yes], [12, 8, 10, 10, Yes], [12, 9, 9, 9, Yes], [12, 10, 8, 8, Yes], [12, 11, 7, 7, Yes], [12, 12, 6, 6, Yes], [12, 13, 5, 5, Yes], [12, 14, 4, 4, Yes], [13, 0, 2, 2, Yes], [13, 1, 1, 1, Yes], [13, 2, 0, 0, Yes], [13, 3, 14, 14, Yes], [13, 4, 13, 13, Yes], [13, 5, 12, 12, Yes], [13, 6, 11, 11, Yes], [13, 7, 10, 10, Yes], [13, 8, 9, 9, Yes], [13, 9, 8, 8, Yes], [13, 10, 7, 7, Yes], [13, 11, 6, 6, Yes], [13, 12, 5, 5, Yes], [13, 13, 4, 4, Yes], [13, 14, 3, 3, Yes], [14, 0, 1, 1, Yes], [14, 1, 0, 0, Yes], [14, 2, 14, 14, Yes], [14, 3, 13, 13, Yes], [14, 4, 12, 12, Yes], [14, 5, 11, 11, Yes], [14, 6, 10, 10, Yes], [14, 7, 9, 9, Yes], [14, 8, 8, 8, Yes], [14, 9, 7, 7, Yes], [14, 10, 6, 6, Yes], [14, 11, 5, 5, Yes], [14, 12, 4, 4, Yes], [14, 13, 3, 3, Yes], [14, 14, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)
List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (15, 15, [0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 14, 14, Yes], [0, 2, 13, 13, Yes], [0, 3, 12, 12, Yes], [0, 4, 11, 11, Yes], [0, 5, 10, 10, Yes], [0, 6, 9, 9, Yes], [0, 7, 8, 8, Yes], [0, 8, 7, 7, Yes], [0, 9, 6, 6, Yes], [0, 10, 5, 5, Yes], [0, 11, 4, 4, Yes], [0, 12, 3, 3, Yes], [0, 13, 2, 2, Yes], [0, 14, 1, 1, Yes], [1, 0, 14, 14, Yes], [1, 1, 13, 13, Yes], [1, 2, 12, 12, Yes], [1, 3, 11, 11, Yes], [1, 4, 10, 10, Yes], [1, 5, 9, 9, Yes], [1, 6, 8, 8, Yes], [1, 7, 7, 7, Yes], [1, 8, 6, 6, Yes], [1, 9, 5, 5, Yes], [1, 10, 4, 4, Yes], [1, 11, 3, 3, Yes], [1, 12, 2, 2, Yes], [1, 13, 1, 1, Yes], [1, 14, 0, 0, Yes], [2, 0, 13, 13, Yes], [2, 1, 12, 12, Yes], [2, 2, 11, 11, Yes], [2, 3, 10, 10, Yes], [2, 4, 9, 9, Yes], [2, 5, 8, 8, Yes], [2, 6, 7, 7, Yes], [2, 7, 6, 6, Yes], [2, 8, 5, 5, Yes], [2, 9, 4, 4, Yes], [2, 10, 3, 3, Yes], [2, 11, 2, 2, Yes], [2, 12, 1, 1, Yes], [2, 13, 0, 0, Yes], [2, 14, 14, 14, Yes], [3, 0, 12, 12, Yes], [3, 1, 11, 11, Yes], [3, 2, 10, 10, Yes], [3, 3, 9, 9, Yes], [3, 4, 8, 8, Yes], [3, 5, 7, 7, Yes], [3, 6, 6, 6, Yes], [3, 7, 5, 5, Yes], [3, 8, 4, 4, Yes], [3, 9, 3, 3, Yes], [3, 10, 2, 2, Yes], [3, 11, 1, 1, Yes], [3, 12, 0, 0, Yes], [3, 13, 14, 14, Yes], [3, 14, 13, 13, Yes], [4, 0, 11, 11, Yes], [4, 1, 10, 10, Yes], [4, 2, 9, 9, Yes], [4, 3, 8, 8, Yes], [4, 4, 7, 7, Yes], [4, 5, 6, 6, Yes], [4, 6, 5, 5, Yes], [4, 7, 4, 4, Yes], [4, 8, 3, 3, Yes], [4, 9, 2, 2, Yes], [4, 10, 1, 1, Yes], [4, 11, 0, 0, Yes], [4, 12, 14, 14, Yes], [4, 13, 13, 13, Yes], [4, 14, 12, 12, Yes], [5, 0, 10, 10, Yes], [5, 1, 9, 9, Yes], [5, 2, 8, 8, Yes], [5, 3, 7, 7, Yes], [5, 4, 6, 6, Yes], [5, 5, 5, 5, Yes], [5, 6, 4, 4, Yes], [5, 7, 3, 3, Yes], [5, 8, 2, 2, Yes], [5, 9, 1, 1, Yes], [5, 10, 0, 0, Yes], [5, 11, 14, 14, Yes], [5, 12, 13, 13, Yes], [5, 13, 12, 12, Yes], [5, 14, 11, 11, Yes], [6, 0, 9, 9, Yes], [6, 1, 8, 8, Yes], [6, 2, 7, 7, Yes], [6, 3, 6, 6, Yes], [6, 4, 5, 5, Yes], [6, 5, 4, 4, Yes], [6, 6, 3, 3, Yes], [6, 7, 2, 2, Yes], [6, 8, 1, 1, Yes], [6, 9, 0, 0, Yes], [6, 10, 14, 14, Yes], [6, 11, 13, 13, Yes], [6, 12, 12, 12, Yes], [6, 13, 11, 11, Yes], [6, 14, 10, 10, Yes], [7, 0, 8, 8, Yes], [7, 1, 7, 7, Yes], [7, 2, 6, 6, Yes], [7, 3, 5, 5, Yes], [7, 4, 4, 4, Yes], [7, 5, 3, 3, Yes], [7, 6, 2, 2, Yes], [7, 7, 1, 1, Yes], [7, 8, 0, 0, Yes], [7, 9, 14, 14, Yes], [7, 10, 13, 13, Yes], [7, 11, 12, 12, Yes], [7, 12, 11, 11, Yes], [7, 13, 10, 10, Yes], [7, 14, 9, 9, Yes], [8, 0, 7, 7, Yes], [8, 1, 6, 6, Yes], [8, 2, 5, 5, Yes], [8, 3, 4, 4, Yes], [8, 4, 3, 3, Yes], [8, 5, 2, 2, Yes], [8, 6, 1, 1, Yes], [8, 7, 0, 0, Yes], [8, 8, 14, 14, Yes], [8, 9, 13, 13, Yes], [8, 10, 12, 12, Yes], [8, 11, 11, 11, Yes], [8, 12, 10, 10, Yes], [8, 13, 9, 9, Yes], [8, 14, 8, 8, Yes], [9, 0, 6, 6, Yes], [9, 1, 5, 5, Yes], [9, 2, 4, 4, Yes], [9, 3, 3, 3, Yes], [9, 4, 2, 2, Yes], [9, 5, 1, 1, Yes], [9, 6, 0, 0, Yes], [9, 7, 14, 14, Yes], [9, 8, 13, 13, Yes], [9, 9, 12, 12, Yes], [9, 10, 11, 11, Yes], [9, 11, 10, 10, Yes], [9, 12, 9, 9, Yes], [9, 13, 8, 8, Yes], [9, 14, 7, 7, Yes], [10, 0, 5, 5, Yes], [10, 1, 4, 4, Yes], [10, 2, 3, 3, Yes], [10, 3, 2, 2, Yes], [10, 4, 1, 1, Yes], [10, 5, 0, 0, Yes], [10, 6, 14, 14, Yes], [10, 7, 13, 13, Yes], [10, 8, 12, 12, Yes], [10, 9, 11, 11, Yes], [10, 10, 10, 10, Yes], [10, 11, 9, 9, Yes], [10, 12, 8, 8, Yes], [10, 13, 7, 7, Yes], [10, 14, 6, 6, Yes], [11, 0, 4, 4, Yes], [11, 1, 3, 3, Yes], [11, 2, 2, 2, Yes], [11, 3, 1, 1, Yes], [11, 4, 0, 0, Yes], [11, 5, 14, 14, Yes], [11, 6, 13, 13, Yes], [11, 7, 12, 12, Yes], [11, 8, 11, 11, Yes], [11, 9, 10, 10, Yes], [11, 10, 9, 9, Yes], [11, 11, 8, 8, Yes], [11, 12, 7, 7, Yes], [11, 13, 6, 6, Yes], [11, 14, 5, 5, Yes], [12, 0, 3, 3, Yes], [12, 1, 2, 2, Yes], [12, 2, 1, 1, Yes], [12, 3, 0, 0, Yes], [12, 4, 14, 14, Yes], [12, 5, 13, 13, Yes], [12, 6, 12, 12, Yes], [12, 7, 11, 11, Yes], [12, 8, 10, 10, Yes], [12, 9, 9, 9, Yes], [12, 10, 8, 8, Yes], [12, 11, 7, 7, Yes], [12, 12, 6, 6, Yes], [12, 13, 5, 5, Yes], [12, 14, 4, 4, Yes], [13, 0, 2, 2, Yes], [13, 1, 1, 1, Yes], [13, 2, 0, 0, Yes], [13, 3, 14, 14, Yes], [13, 4, 13, 13, Yes], [13, 5, 12, 12, Yes], [13, 6, 11, 11, Yes], [13, 7, 10, 10, Yes], [13, 8, 9, 9, Yes], [13, 9, 8, 8, Yes], [13, 10, 7, 7, Yes], [13, 11, 6, 6, Yes], [13, 12, 5, 5, Yes], [13, 13, 4, 4, Yes], [13, 14, 3, 3, Yes], [14, 0, 1, 1, Yes], [14, 1, 0, 0, Yes], [14, 2, 14, 14, Yes], [14, 3, 13, 13, Yes], [14, 4, 12, 12, Yes], [14, 5, 11, 11, Yes], [14, 6, 10, 10, Yes], [14, 7, 9, 9, Yes], [14, 8, 8, 8, Yes], [14, 9, 7, 7, Yes], [14, 10, 6, 6, Yes], [14, 11, 5, 5, Yes], [14, 12, 4, 4, Yes], [14, 13, 3, 3, Yes], [14, 14, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-homomorphism-qualification · candidate
14. Qualify Both Groups and Every Product Pair before Applying Kernel Theorems
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]], target identity=0, and total mapping=[0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).
View answer and explanation
(16, 16, [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 15, 15, Yes], [0, 2, 14, 14, Yes], [0, 3, 13, 13, Yes], [0, 4, 12, 12, Yes], [0, 5, 11, 11, Yes], [0, 6, 10, 10, Yes], [0, 7, 9, 9, Yes], [0, 8, 8, 8, Yes], [0, 9, 7, 7, Yes], [0, 10, 6, 6, Yes], [0, 11, 5, 5, Yes], [0, 12, 4, 4, Yes], [0, 13, 3, 3, Yes], [0, 14, 2, 2, Yes], [0, 15, 1, 1, Yes], [1, 0, 15, 15, Yes], [1, 1, 14, 14, Yes], [1, 2, 13, 13, Yes], [1, 3, 12, 12, Yes], [1, 4, 11, 11, Yes], [1, 5, 10, 10, Yes], [1, 6, 9, 9, Yes], [1, 7, 8, 8, Yes], [1, 8, 7, 7, Yes], [1, 9, 6, 6, Yes], [1, 10, 5, 5, Yes], [1, 11, 4, 4, Yes], [1, 12, 3, 3, Yes], [1, 13, 2, 2, Yes], [1, 14, 1, 1, Yes], [1, 15, 0, 0, Yes], [2, 0, 14, 14, Yes], [2, 1, 13, 13, Yes], [2, 2, 12, 12, Yes], [2, 3, 11, 11, Yes], [2, 4, 10, 10, Yes], [2, 5, 9, 9, Yes], [2, 6, 8, 8, Yes], [2, 7, 7, 7, Yes], [2, 8, 6, 6, Yes], [2, 9, 5, 5, Yes], [2, 10, 4, 4, Yes], [2, 11, 3, 3, Yes], [2, 12, 2, 2, Yes], [2, 13, 1, 1, Yes], [2, 14, 0, 0, Yes], [2, 15, 15, 15, Yes], [3, 0, 13, 13, Yes], [3, 1, 12, 12, Yes], [3, 2, 11, 11, Yes], [3, 3, 10, 10, Yes], [3, 4, 9, 9, Yes], [3, 5, 8, 8, Yes], [3, 6, 7, 7, Yes], [3, 7, 6, 6, Yes], [3, 8, 5, 5, Yes], [3, 9, 4, 4, Yes], [3, 10, 3, 3, Yes], [3, 11, 2, 2, Yes], [3, 12, 1, 1, Yes], [3, 13, 0, 0, Yes], [3, 14, 15, 15, Yes], [3, 15, 14, 14, Yes], [4, 0, 12, 12, Yes], [4, 1, 11, 11, Yes], [4, 2, 10, 10, Yes], [4, 3, 9, 9, Yes], [4, 4, 8, 8, Yes], [4, 5, 7, 7, Yes], [4, 6, 6, 6, Yes], [4, 7, 5, 5, Yes], [4, 8, 4, 4, Yes], [4, 9, 3, 3, Yes], [4, 10, 2, 2, Yes], [4, 11, 1, 1, Yes], [4, 12, 0, 0, Yes], [4, 13, 15, 15, Yes], [4, 14, 14, 14, Yes], [4, 15, 13, 13, Yes], [5, 0, 11, 11, Yes], [5, 1, 10, 10, Yes], [5, 2, 9, 9, Yes], [5, 3, 8, 8, Yes], [5, 4, 7, 7, Yes], [5, 5, 6, 6, Yes], [5, 6, 5, 5, Yes], [5, 7, 4, 4, Yes], [5, 8, 3, 3, Yes], [5, 9, 2, 2, Yes], [5, 10, 1, 1, Yes], [5, 11, 0, 0, Yes], [5, 12, 15, 15, Yes], [5, 13, 14, 14, Yes], [5, 14, 13, 13, Yes], [5, 15, 12, 12, Yes], [6, 0, 10, 10, Yes], [6, 1, 9, 9, Yes], [6, 2, 8, 8, Yes], [6, 3, 7, 7, Yes], [6, 4, 6, 6, Yes], [6, 5, 5, 5, Yes], [6, 6, 4, 4, Yes], [6, 7, 3, 3, Yes], [6, 8, 2, 2, Yes], [6, 9, 1, 1, Yes], [6, 10, 0, 0, Yes], [6, 11, 15, 15, Yes], [6, 12, 14, 14, Yes], [6, 13, 13, 13, Yes], [6, 14, 12, 12, Yes], [6, 15, 11, 11, Yes], [7, 0, 9, 9, Yes], [7, 1, 8, 8, Yes], [7, 2, 7, 7, Yes], [7, 3, 6, 6, Yes], [7, 4, 5, 5, Yes], [7, 5, 4, 4, Yes], [7, 6, 3, 3, Yes], [7, 7, 2, 2, Yes], [7, 8, 1, 1, Yes], [7, 9, 0, 0, Yes], [7, 10, 15, 15, Yes], [7, 11, 14, 14, Yes], [7, 12, 13, 13, Yes], [7, 13, 12, 12, Yes], [7, 14, 11, 11, Yes], [7, 15, 10, 10, Yes], [8, 0, 8, 8, Yes], [8, 1, 7, 7, Yes], [8, 2, 6, 6, Yes], [8, 3, 5, 5, Yes], [8, 4, 4, 4, Yes], [8, 5, 3, 3, Yes], [8, 6, 2, 2, Yes], [8, 7, 1, 1, Yes], [8, 8, 0, 0, Yes], [8, 9, 15, 15, Yes], [8, 10, 14, 14, Yes], [8, 11, 13, 13, Yes], [8, 12, 12, 12, Yes], [8, 13, 11, 11, Yes], [8, 14, 10, 10, Yes], [8, 15, 9, 9, Yes], [9, 0, 7, 7, Yes], [9, 1, 6, 6, Yes], [9, 2, 5, 5, Yes], [9, 3, 4, 4, Yes], [9, 4, 3, 3, Yes], [9, 5, 2, 2, Yes], [9, 6, 1, 1, Yes], [9, 7, 0, 0, Yes], [9, 8, 15, 15, Yes], [9, 9, 14, 14, Yes], [9, 10, 13, 13, Yes], [9, 11, 12, 12, Yes], [9, 12, 11, 11, Yes], [9, 13, 10, 10, Yes], [9, 14, 9, 9, Yes], [9, 15, 8, 8, Yes], [10, 0, 6, 6, Yes], [10, 1, 5, 5, Yes], [10, 2, 4, 4, Yes], [10, 3, 3, 3, Yes], [10, 4, 2, 2, Yes], [10, 5, 1, 1, Yes], [10, 6, 0, 0, Yes], [10, 7, 15, 15, Yes], [10, 8, 14, 14, Yes], [10, 9, 13, 13, Yes], [10, 10, 12, 12, Yes], [10, 11, 11, 11, Yes], [10, 12, 10, 10, Yes], [10, 13, 9, 9, Yes], [10, 14, 8, 8, Yes], [10, 15, 7, 7, Yes], [11, 0, 5, 5, Yes], [11, 1, 4, 4, Yes], [11, 2, 3, 3, Yes], [11, 3, 2, 2, Yes], [11, 4, 1, 1, Yes], [11, 5, 0, 0, Yes], [11, 6, 15, 15, Yes], [11, 7, 14, 14, Yes], [11, 8, 13, 13, Yes], [11, 9, 12, 12, Yes], [11, 10, 11, 11, Yes], [11, 11, 10, 10, Yes], [11, 12, 9, 9, Yes], [11, 13, 8, 8, Yes], [11, 14, 7, 7, Yes], [11, 15, 6, 6, Yes], [12, 0, 4, 4, Yes], [12, 1, 3, 3, Yes], [12, 2, 2, 2, Yes], [12, 3, 1, 1, Yes], [12, 4, 0, 0, Yes], [12, 5, 15, 15, Yes], [12, 6, 14, 14, Yes], [12, 7, 13, 13, Yes], [12, 8, 12, 12, Yes], [12, 9, 11, 11, Yes], [12, 10, 10, 10, Yes], [12, 11, 9, 9, Yes], [12, 12, 8, 8, Yes], [12, 13, 7, 7, Yes], [12, 14, 6, 6, Yes], [12, 15, 5, 5, Yes], [13, 0, 3, 3, Yes], [13, 1, 2, 2, Yes], [13, 2, 1, 1, Yes], [13, 3, 0, 0, Yes], [13, 4, 15, 15, Yes], [13, 5, 14, 14, Yes], [13, 6, 13, 13, Yes], [13, 7, 12, 12, Yes], [13, 8, 11, 11, Yes], [13, 9, 10, 10, Yes], [13, 10, 9, 9, Yes], [13, 11, 8, 8, Yes], [13, 12, 7, 7, Yes], [13, 13, 6, 6, Yes], [13, 14, 5, 5, Yes], [13, 15, 4, 4, Yes], [14, 0, 2, 2, Yes], [14, 1, 1, 1, Yes], [14, 2, 0, 0, Yes], [14, 3, 15, 15, Yes], [14, 4, 14, 14, Yes], [14, 5, 13, 13, Yes], [14, 6, 12, 12, Yes], [14, 7, 11, 11, Yes], [14, 8, 10, 10, Yes], [14, 9, 9, 9, Yes], [14, 10, 8, 8, Yes], [14, 11, 7, 7, Yes], [14, 12, 6, 6, Yes], [14, 13, 5, 5, Yes], [14, 14, 4, 4, Yes], [14, 15, 3, 3, Yes], [15, 0, 1, 1, Yes], [15, 1, 0, 0, Yes], [15, 2, 15, 15, Yes], [15, 3, 14, 14, Yes], [15, 4, 13, 13, Yes], [15, 5, 12, 12, Yes], [15, 6, 11, 11, Yes], [15, 7, 10, 10, Yes], [15, 8, 9, 9, Yes], [15, 9, 8, 8, Yes], [15, 10, 7, 7, Yes], [15, 11, 6, 6, Yes], [15, 12, 5, 5, Yes], [15, 13, 4, 4, Yes], [15, 14, 3, 3, Yes], [15, 15, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)
List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (16, 16, [0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 15, 15, Yes], [0, 2, 14, 14, Yes], [0, 3, 13, 13, Yes], [0, 4, 12, 12, Yes], [0, 5, 11, 11, Yes], [0, 6, 10, 10, Yes], [0, 7, 9, 9, Yes], [0, 8, 8, 8, Yes], [0, 9, 7, 7, Yes], [0, 10, 6, 6, Yes], [0, 11, 5, 5, Yes], [0, 12, 4, 4, Yes], [0, 13, 3, 3, Yes], [0, 14, 2, 2, Yes], [0, 15, 1, 1, Yes], [1, 0, 15, 15, Yes], [1, 1, 14, 14, Yes], [1, 2, 13, 13, Yes], [1, 3, 12, 12, Yes], [1, 4, 11, 11, Yes], [1, 5, 10, 10, Yes], [1, 6, 9, 9, Yes], [1, 7, 8, 8, Yes], [1, 8, 7, 7, Yes], [1, 9, 6, 6, Yes], [1, 10, 5, 5, Yes], [1, 11, 4, 4, Yes], [1, 12, 3, 3, Yes], [1, 13, 2, 2, Yes], [1, 14, 1, 1, Yes], [1, 15, 0, 0, Yes], [2, 0, 14, 14, Yes], [2, 1, 13, 13, Yes], [2, 2, 12, 12, Yes], [2, 3, 11, 11, Yes], [2, 4, 10, 10, Yes], [2, 5, 9, 9, Yes], [2, 6, 8, 8, Yes], [2, 7, 7, 7, Yes], [2, 8, 6, 6, Yes], [2, 9, 5, 5, Yes], [2, 10, 4, 4, Yes], [2, 11, 3, 3, Yes], [2, 12, 2, 2, Yes], [2, 13, 1, 1, Yes], [2, 14, 0, 0, Yes], [2, 15, 15, 15, Yes], [3, 0, 13, 13, Yes], [3, 1, 12, 12, Yes], [3, 2, 11, 11, Yes], [3, 3, 10, 10, Yes], [3, 4, 9, 9, Yes], [3, 5, 8, 8, Yes], [3, 6, 7, 7, Yes], [3, 7, 6, 6, Yes], [3, 8, 5, 5, Yes], [3, 9, 4, 4, Yes], [3, 10, 3, 3, Yes], [3, 11, 2, 2, Yes], [3, 12, 1, 1, Yes], [3, 13, 0, 0, Yes], [3, 14, 15, 15, Yes], [3, 15, 14, 14, Yes], [4, 0, 12, 12, Yes], [4, 1, 11, 11, Yes], [4, 2, 10, 10, Yes], [4, 3, 9, 9, Yes], [4, 4, 8, 8, Yes], [4, 5, 7, 7, Yes], [4, 6, 6, 6, Yes], [4, 7, 5, 5, Yes], [4, 8, 4, 4, Yes], [4, 9, 3, 3, Yes], [4, 10, 2, 2, Yes], [4, 11, 1, 1, Yes], [4, 12, 0, 0, Yes], [4, 13, 15, 15, Yes], [4, 14, 14, 14, Yes], [4, 15, 13, 13, Yes], [5, 0, 11, 11, Yes], [5, 1, 10, 10, Yes], [5, 2, 9, 9, Yes], [5, 3, 8, 8, Yes], [5, 4, 7, 7, Yes], [5, 5, 6, 6, Yes], [5, 6, 5, 5, Yes], [5, 7, 4, 4, Yes], [5, 8, 3, 3, Yes], [5, 9, 2, 2, Yes], [5, 10, 1, 1, Yes], [5, 11, 0, 0, Yes], [5, 12, 15, 15, Yes], [5, 13, 14, 14, Yes], [5, 14, 13, 13, Yes], [5, 15, 12, 12, Yes], [6, 0, 10, 10, Yes], [6, 1, 9, 9, Yes], [6, 2, 8, 8, Yes], [6, 3, 7, 7, Yes], [6, 4, 6, 6, Yes], [6, 5, 5, 5, Yes], [6, 6, 4, 4, Yes], [6, 7, 3, 3, Yes], [6, 8, 2, 2, Yes], [6, 9, 1, 1, Yes], [6, 10, 0, 0, Yes], [6, 11, 15, 15, Yes], [6, 12, 14, 14, Yes], [6, 13, 13, 13, Yes], [6, 14, 12, 12, Yes], [6, 15, 11, 11, Yes], [7, 0, 9, 9, Yes], [7, 1, 8, 8, Yes], [7, 2, 7, 7, Yes], [7, 3, 6, 6, Yes], [7, 4, 5, 5, Yes], [7, 5, 4, 4, Yes], [7, 6, 3, 3, Yes], [7, 7, 2, 2, Yes], [7, 8, 1, 1, Yes], [7, 9, 0, 0, Yes], [7, 10, 15, 15, Yes], [7, 11, 14, 14, Yes], [7, 12, 13, 13, Yes], [7, 13, 12, 12, Yes], [7, 14, 11, 11, Yes], [7, 15, 10, 10, Yes], [8, 0, 8, 8, Yes], [8, 1, 7, 7, Yes], [8, 2, 6, 6, Yes], [8, 3, 5, 5, Yes], [8, 4, 4, 4, Yes], [8, 5, 3, 3, Yes], [8, 6, 2, 2, Yes], [8, 7, 1, 1, Yes], [8, 8, 0, 0, Yes], [8, 9, 15, 15, Yes], [8, 10, 14, 14, Yes], [8, 11, 13, 13, Yes], [8, 12, 12, 12, Yes], [8, 13, 11, 11, Yes], [8, 14, 10, 10, Yes], [8, 15, 9, 9, Yes], [9, 0, 7, 7, Yes], [9, 1, 6, 6, Yes], [9, 2, 5, 5, Yes], [9, 3, 4, 4, Yes], [9, 4, 3, 3, Yes], [9, 5, 2, 2, Yes], [9, 6, 1, 1, Yes], [9, 7, 0, 0, Yes], [9, 8, 15, 15, Yes], [9, 9, 14, 14, Yes], [9, 10, 13, 13, Yes], [9, 11, 12, 12, Yes], [9, 12, 11, 11, Yes], [9, 13, 10, 10, Yes], [9, 14, 9, 9, Yes], [9, 15, 8, 8, Yes], [10, 0, 6, 6, Yes], [10, 1, 5, 5, Yes], [10, 2, 4, 4, Yes], [10, 3, 3, 3, Yes], [10, 4, 2, 2, Yes], [10, 5, 1, 1, Yes], [10, 6, 0, 0, Yes], [10, 7, 15, 15, Yes], [10, 8, 14, 14, Yes], [10, 9, 13, 13, Yes], [10, 10, 12, 12, Yes], [10, 11, 11, 11, Yes], [10, 12, 10, 10, Yes], [10, 13, 9, 9, Yes], [10, 14, 8, 8, Yes], [10, 15, 7, 7, Yes], [11, 0, 5, 5, Yes], [11, 1, 4, 4, Yes], [11, 2, 3, 3, Yes], [11, 3, 2, 2, Yes], [11, 4, 1, 1, Yes], [11, 5, 0, 0, Yes], [11, 6, 15, 15, Yes], [11, 7, 14, 14, Yes], [11, 8, 13, 13, Yes], [11, 9, 12, 12, Yes], [11, 10, 11, 11, Yes], [11, 11, 10, 10, Yes], [11, 12, 9, 9, Yes], [11, 13, 8, 8, Yes], [11, 14, 7, 7, Yes], [11, 15, 6, 6, Yes], [12, 0, 4, 4, Yes], [12, 1, 3, 3, Yes], [12, 2, 2, 2, Yes], [12, 3, 1, 1, Yes], [12, 4, 0, 0, Yes], [12, 5, 15, 15, Yes], [12, 6, 14, 14, Yes], [12, 7, 13, 13, Yes], [12, 8, 12, 12, Yes], [12, 9, 11, 11, Yes], [12, 10, 10, 10, Yes], [12, 11, 9, 9, Yes], [12, 12, 8, 8, Yes], [12, 13, 7, 7, Yes], [12, 14, 6, 6, Yes], [12, 15, 5, 5, Yes], [13, 0, 3, 3, Yes], [13, 1, 2, 2, Yes], [13, 2, 1, 1, Yes], [13, 3, 0, 0, Yes], [13, 4, 15, 15, Yes], [13, 5, 14, 14, Yes], [13, 6, 13, 13, Yes], [13, 7, 12, 12, Yes], [13, 8, 11, 11, Yes], [13, 9, 10, 10, Yes], [13, 10, 9, 9, Yes], [13, 11, 8, 8, Yes], [13, 12, 7, 7, Yes], [13, 13, 6, 6, Yes], [13, 14, 5, 5, Yes], [13, 15, 4, 4, Yes], [14, 0, 2, 2, Yes], [14, 1, 1, 1, Yes], [14, 2, 0, 0, Yes], [14, 3, 15, 15, Yes], [14, 4, 14, 14, Yes], [14, 5, 13, 13, Yes], [14, 6, 12, 12, Yes], [14, 7, 11, 11, Yes], [14, 8, 10, 10, Yes], [14, 9, 9, 9, Yes], [14, 10, 8, 8, Yes], [14, 11, 7, 7, Yes], [14, 12, 6, 6, Yes], [14, 13, 5, 5, Yes], [14, 14, 4, 4, Yes], [14, 15, 3, 3, Yes], [15, 0, 1, 1, Yes], [15, 1, 0, 0, Yes], [15, 2, 15, 15, Yes], [15, 3, 14, 14, Yes], [15, 4, 13, 13, Yes], [15, 5, 12, 12, Yes], [15, 6, 11, 11, Yes], [15, 7, 10, 10, Yes], [15, 8, 9, 9, Yes], [15, 9, 8, 8, Yes], [15, 10, 7, 7, Yes], [15, 11, 6, 6, Yes], [15, 12, 5, 5, Yes], [15, 13, 4, 4, Yes], [15, 14, 3, 3, Yes], [15, 15, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-homomorphism-qualification · candidate
15. Qualify Both Groups and Every Product Pair before Applying Kernel Theorems
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]], target identity=0, and total mapping=[0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).
View answer and explanation
(17, 17, [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 16, 16, Yes], [0, 2, 15, 15, Yes], [0, 3, 14, 14, Yes], [0, 4, 13, 13, Yes], [0, 5, 12, 12, Yes], [0, 6, 11, 11, Yes], [0, 7, 10, 10, Yes], [0, 8, 9, 9, Yes], [0, 9, 8, 8, Yes], [0, 10, 7, 7, Yes], [0, 11, 6, 6, Yes], [0, 12, 5, 5, Yes], [0, 13, 4, 4, Yes], [0, 14, 3, 3, Yes], [0, 15, 2, 2, Yes], [0, 16, 1, 1, Yes], [1, 0, 16, 16, Yes], [1, 1, 15, 15, Yes], [1, 2, 14, 14, Yes], [1, 3, 13, 13, Yes], [1, 4, 12, 12, Yes], [1, 5, 11, 11, Yes], [1, 6, 10, 10, Yes], [1, 7, 9, 9, Yes], [1, 8, 8, 8, Yes], [1, 9, 7, 7, Yes], [1, 10, 6, 6, Yes], [1, 11, 5, 5, Yes], [1, 12, 4, 4, Yes], [1, 13, 3, 3, Yes], [1, 14, 2, 2, Yes], [1, 15, 1, 1, Yes], [1, 16, 0, 0, Yes], [2, 0, 15, 15, Yes], [2, 1, 14, 14, Yes], [2, 2, 13, 13, Yes], [2, 3, 12, 12, Yes], [2, 4, 11, 11, Yes], [2, 5, 10, 10, Yes], [2, 6, 9, 9, Yes], [2, 7, 8, 8, Yes], [2, 8, 7, 7, Yes], [2, 9, 6, 6, Yes], [2, 10, 5, 5, Yes], [2, 11, 4, 4, Yes], [2, 12, 3, 3, Yes], [2, 13, 2, 2, Yes], [2, 14, 1, 1, Yes], [2, 15, 0, 0, Yes], [2, 16, 16, 16, Yes], [3, 0, 14, 14, Yes], [3, 1, 13, 13, Yes], [3, 2, 12, 12, Yes], [3, 3, 11, 11, Yes], [3, 4, 10, 10, Yes], [3, 5, 9, 9, Yes], [3, 6, 8, 8, Yes], [3, 7, 7, 7, Yes], [3, 8, 6, 6, Yes], [3, 9, 5, 5, Yes], [3, 10, 4, 4, Yes], [3, 11, 3, 3, Yes], [3, 12, 2, 2, Yes], [3, 13, 1, 1, Yes], [3, 14, 0, 0, Yes], [3, 15, 16, 16, Yes], [3, 16, 15, 15, Yes], [4, 0, 13, 13, Yes], [4, 1, 12, 12, Yes], [4, 2, 11, 11, Yes], [4, 3, 10, 10, Yes], [4, 4, 9, 9, Yes], [4, 5, 8, 8, Yes], [4, 6, 7, 7, Yes], [4, 7, 6, 6, Yes], [4, 8, 5, 5, Yes], [4, 9, 4, 4, Yes], [4, 10, 3, 3, Yes], [4, 11, 2, 2, Yes], [4, 12, 1, 1, Yes], [4, 13, 0, 0, Yes], [4, 14, 16, 16, Yes], [4, 15, 15, 15, Yes], [4, 16, 14, 14, Yes], [5, 0, 12, 12, Yes], [5, 1, 11, 11, Yes], [5, 2, 10, 10, Yes], [5, 3, 9, 9, Yes], [5, 4, 8, 8, Yes], [5, 5, 7, 7, Yes], [5, 6, 6, 6, Yes], [5, 7, 5, 5, Yes], [5, 8, 4, 4, Yes], [5, 9, 3, 3, Yes], [5, 10, 2, 2, Yes], [5, 11, 1, 1, Yes], [5, 12, 0, 0, Yes], [5, 13, 16, 16, Yes], [5, 14, 15, 15, Yes], [5, 15, 14, 14, Yes], [5, 16, 13, 13, Yes], [6, 0, 11, 11, Yes], [6, 1, 10, 10, Yes], [6, 2, 9, 9, Yes], [6, 3, 8, 8, Yes], [6, 4, 7, 7, Yes], [6, 5, 6, 6, Yes], [6, 6, 5, 5, Yes], [6, 7, 4, 4, Yes], [6, 8, 3, 3, Yes], [6, 9, 2, 2, Yes], [6, 10, 1, 1, Yes], [6, 11, 0, 0, Yes], [6, 12, 16, 16, Yes], [6, 13, 15, 15, Yes], [6, 14, 14, 14, Yes], [6, 15, 13, 13, Yes], [6, 16, 12, 12, Yes], [7, 0, 10, 10, Yes], [7, 1, 9, 9, Yes], [7, 2, 8, 8, Yes], [7, 3, 7, 7, Yes], [7, 4, 6, 6, Yes], [7, 5, 5, 5, Yes], [7, 6, 4, 4, Yes], [7, 7, 3, 3, Yes], [7, 8, 2, 2, Yes], [7, 9, 1, 1, Yes], [7, 10, 0, 0, Yes], [7, 11, 16, 16, Yes], [7, 12, 15, 15, Yes], [7, 13, 14, 14, Yes], [7, 14, 13, 13, Yes], [7, 15, 12, 12, Yes], [7, 16, 11, 11, Yes], [8, 0, 9, 9, Yes], [8, 1, 8, 8, Yes], [8, 2, 7, 7, Yes], [8, 3, 6, 6, Yes], [8, 4, 5, 5, Yes], [8, 5, 4, 4, Yes], [8, 6, 3, 3, Yes], [8, 7, 2, 2, Yes], [8, 8, 1, 1, Yes], [8, 9, 0, 0, Yes], [8, 10, 16, 16, Yes], [8, 11, 15, 15, Yes], [8, 12, 14, 14, Yes], [8, 13, 13, 13, Yes], [8, 14, 12, 12, Yes], [8, 15, 11, 11, Yes], [8, 16, 10, 10, Yes], [9, 0, 8, 8, Yes], [9, 1, 7, 7, Yes], [9, 2, 6, 6, Yes], [9, 3, 5, 5, Yes], [9, 4, 4, 4, Yes], [9, 5, 3, 3, Yes], [9, 6, 2, 2, Yes], [9, 7, 1, 1, Yes], [9, 8, 0, 0, Yes], [9, 9, 16, 16, Yes], [9, 10, 15, 15, Yes], [9, 11, 14, 14, Yes], [9, 12, 13, 13, Yes], [9, 13, 12, 12, Yes], [9, 14, 11, 11, Yes], [9, 15, 10, 10, Yes], [9, 16, 9, 9, Yes], [10, 0, 7, 7, Yes], [10, 1, 6, 6, Yes], [10, 2, 5, 5, Yes], [10, 3, 4, 4, Yes], [10, 4, 3, 3, Yes], [10, 5, 2, 2, Yes], [10, 6, 1, 1, Yes], [10, 7, 0, 0, Yes], [10, 8, 16, 16, Yes], [10, 9, 15, 15, Yes], [10, 10, 14, 14, Yes], [10, 11, 13, 13, Yes], [10, 12, 12, 12, Yes], [10, 13, 11, 11, Yes], [10, 14, 10, 10, Yes], [10, 15, 9, 9, Yes], [10, 16, 8, 8, Yes], [11, 0, 6, 6, Yes], [11, 1, 5, 5, Yes], [11, 2, 4, 4, Yes], [11, 3, 3, 3, Yes], [11, 4, 2, 2, Yes], [11, 5, 1, 1, Yes], [11, 6, 0, 0, Yes], [11, 7, 16, 16, Yes], [11, 8, 15, 15, Yes], [11, 9, 14, 14, Yes], [11, 10, 13, 13, Yes], [11, 11, 12, 12, Yes], [11, 12, 11, 11, Yes], [11, 13, 10, 10, Yes], [11, 14, 9, 9, Yes], [11, 15, 8, 8, Yes], [11, 16, 7, 7, Yes], [12, 0, 5, 5, Yes], [12, 1, 4, 4, Yes], [12, 2, 3, 3, Yes], [12, 3, 2, 2, Yes], [12, 4, 1, 1, Yes], [12, 5, 0, 0, Yes], [12, 6, 16, 16, Yes], [12, 7, 15, 15, Yes], [12, 8, 14, 14, Yes], [12, 9, 13, 13, Yes], [12, 10, 12, 12, Yes], [12, 11, 11, 11, Yes], [12, 12, 10, 10, Yes], [12, 13, 9, 9, Yes], [12, 14, 8, 8, Yes], [12, 15, 7, 7, Yes], [12, 16, 6, 6, Yes], [13, 0, 4, 4, Yes], [13, 1, 3, 3, Yes], [13, 2, 2, 2, Yes], [13, 3, 1, 1, Yes], [13, 4, 0, 0, Yes], [13, 5, 16, 16, Yes], [13, 6, 15, 15, Yes], [13, 7, 14, 14, Yes], [13, 8, 13, 13, Yes], [13, 9, 12, 12, Yes], [13, 10, 11, 11, Yes], [13, 11, 10, 10, Yes], [13, 12, 9, 9, Yes], [13, 13, 8, 8, Yes], [13, 14, 7, 7, Yes], [13, 15, 6, 6, Yes], [13, 16, 5, 5, Yes], [14, 0, 3, 3, Yes], [14, 1, 2, 2, Yes], [14, 2, 1, 1, Yes], [14, 3, 0, 0, Yes], [14, 4, 16, 16, Yes], [14, 5, 15, 15, Yes], [14, 6, 14, 14, Yes], [14, 7, 13, 13, Yes], [14, 8, 12, 12, Yes], [14, 9, 11, 11, Yes], [14, 10, 10, 10, Yes], [14, 11, 9, 9, Yes], [14, 12, 8, 8, Yes], [14, 13, 7, 7, Yes], [14, 14, 6, 6, Yes], [14, 15, 5, 5, Yes], [14, 16, 4, 4, Yes], [15, 0, 2, 2, Yes], [15, 1, 1, 1, Yes], [15, 2, 0, 0, Yes], [15, 3, 16, 16, Yes], [15, 4, 15, 15, Yes], [15, 5, 14, 14, Yes], [15, 6, 13, 13, Yes], [15, 7, 12, 12, Yes], [15, 8, 11, 11, Yes], [15, 9, 10, 10, Yes], [15, 10, 9, 9, Yes], [15, 11, 8, 8, Yes], [15, 12, 7, 7, Yes], [15, 13, 6, 6, Yes], [15, 14, 5, 5, Yes], [15, 15, 4, 4, Yes], [15, 16, 3, 3, Yes], [16, 0, 1, 1, Yes], [16, 1, 0, 0, Yes], [16, 2, 16, 16, Yes], [16, 3, 15, 15, Yes], [16, 4, 14, 14, Yes], [16, 5, 13, 13, Yes], [16, 6, 12, 12, Yes], [16, 7, 11, 11, Yes], [16, 8, 10, 10, Yes], [16, 9, 9, 9, Yes], [16, 10, 8, 8, Yes], [16, 11, 7, 7, Yes], [16, 12, 6, 6, Yes], [16, 13, 5, 5, Yes], [16, 14, 4, 4, Yes], [16, 15, 3, 3, Yes], [16, 16, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)
List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (17, 17, [0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 16, 16, Yes], [0, 2, 15, 15, Yes], [0, 3, 14, 14, Yes], [0, 4, 13, 13, Yes], [0, 5, 12, 12, Yes], [0, 6, 11, 11, Yes], [0, 7, 10, 10, Yes], [0, 8, 9, 9, Yes], [0, 9, 8, 8, Yes], [0, 10, 7, 7, Yes], [0, 11, 6, 6, Yes], [0, 12, 5, 5, Yes], [0, 13, 4, 4, Yes], [0, 14, 3, 3, Yes], [0, 15, 2, 2, Yes], [0, 16, 1, 1, Yes], [1, 0, 16, 16, Yes], [1, 1, 15, 15, Yes], [1, 2, 14, 14, Yes], [1, 3, 13, 13, Yes], [1, 4, 12, 12, Yes], [1, 5, 11, 11, Yes], [1, 6, 10, 10, Yes], [1, 7, 9, 9, Yes], [1, 8, 8, 8, Yes], [1, 9, 7, 7, Yes], [1, 10, 6, 6, Yes], [1, 11, 5, 5, Yes], [1, 12, 4, 4, Yes], [1, 13, 3, 3, Yes], [1, 14, 2, 2, Yes], [1, 15, 1, 1, Yes], [1, 16, 0, 0, Yes], [2, 0, 15, 15, Yes], [2, 1, 14, 14, Yes], [2, 2, 13, 13, Yes], [2, 3, 12, 12, Yes], [2, 4, 11, 11, Yes], [2, 5, 10, 10, Yes], [2, 6, 9, 9, Yes], [2, 7, 8, 8, Yes], [2, 8, 7, 7, Yes], [2, 9, 6, 6, Yes], [2, 10, 5, 5, Yes], [2, 11, 4, 4, Yes], [2, 12, 3, 3, Yes], [2, 13, 2, 2, Yes], [2, 14, 1, 1, Yes], [2, 15, 0, 0, Yes], [2, 16, 16, 16, Yes], [3, 0, 14, 14, Yes], [3, 1, 13, 13, Yes], [3, 2, 12, 12, Yes], [3, 3, 11, 11, Yes], [3, 4, 10, 10, Yes], [3, 5, 9, 9, Yes], [3, 6, 8, 8, Yes], [3, 7, 7, 7, Yes], [3, 8, 6, 6, Yes], [3, 9, 5, 5, Yes], [3, 10, 4, 4, Yes], [3, 11, 3, 3, Yes], [3, 12, 2, 2, Yes], [3, 13, 1, 1, Yes], [3, 14, 0, 0, Yes], [3, 15, 16, 16, Yes], [3, 16, 15, 15, Yes], [4, 0, 13, 13, Yes], [4, 1, 12, 12, Yes], [4, 2, 11, 11, Yes], [4, 3, 10, 10, Yes], [4, 4, 9, 9, Yes], [4, 5, 8, 8, Yes], [4, 6, 7, 7, Yes], [4, 7, 6, 6, Yes], [4, 8, 5, 5, Yes], [4, 9, 4, 4, Yes], [4, 10, 3, 3, Yes], [4, 11, 2, 2, Yes], [4, 12, 1, 1, Yes], [4, 13, 0, 0, Yes], [4, 14, 16, 16, Yes], [4, 15, 15, 15, Yes], [4, 16, 14, 14, Yes], [5, 0, 12, 12, Yes], [5, 1, 11, 11, Yes], [5, 2, 10, 10, Yes], [5, 3, 9, 9, Yes], [5, 4, 8, 8, Yes], [5, 5, 7, 7, Yes], [5, 6, 6, 6, Yes], [5, 7, 5, 5, Yes], [5, 8, 4, 4, Yes], [5, 9, 3, 3, Yes], [5, 10, 2, 2, Yes], [5, 11, 1, 1, Yes], [5, 12, 0, 0, Yes], [5, 13, 16, 16, Yes], [5, 14, 15, 15, Yes], [5, 15, 14, 14, Yes], [5, 16, 13, 13, Yes], [6, 0, 11, 11, Yes], [6, 1, 10, 10, Yes], [6, 2, 9, 9, Yes], [6, 3, 8, 8, Yes], [6, 4, 7, 7, Yes], [6, 5, 6, 6, Yes], [6, 6, 5, 5, Yes], [6, 7, 4, 4, Yes], [6, 8, 3, 3, Yes], [6, 9, 2, 2, Yes], [6, 10, 1, 1, Yes], [6, 11, 0, 0, Yes], [6, 12, 16, 16, Yes], [6, 13, 15, 15, Yes], [6, 14, 14, 14, Yes], [6, 15, 13, 13, Yes], [6, 16, 12, 12, Yes], [7, 0, 10, 10, Yes], [7, 1, 9, 9, Yes], [7, 2, 8, 8, Yes], [7, 3, 7, 7, Yes], [7, 4, 6, 6, Yes], [7, 5, 5, 5, Yes], [7, 6, 4, 4, Yes], [7, 7, 3, 3, Yes], [7, 8, 2, 2, Yes], [7, 9, 1, 1, Yes], [7, 10, 0, 0, Yes], [7, 11, 16, 16, Yes], [7, 12, 15, 15, Yes], [7, 13, 14, 14, Yes], [7, 14, 13, 13, Yes], [7, 15, 12, 12, Yes], [7, 16, 11, 11, Yes], [8, 0, 9, 9, Yes], [8, 1, 8, 8, Yes], [8, 2, 7, 7, Yes], [8, 3, 6, 6, Yes], [8, 4, 5, 5, Yes], [8, 5, 4, 4, Yes], [8, 6, 3, 3, Yes], [8, 7, 2, 2, Yes], [8, 8, 1, 1, Yes], [8, 9, 0, 0, Yes], [8, 10, 16, 16, Yes], [8, 11, 15, 15, Yes], [8, 12, 14, 14, Yes], [8, 13, 13, 13, Yes], [8, 14, 12, 12, Yes], [8, 15, 11, 11, Yes], [8, 16, 10, 10, Yes], [9, 0, 8, 8, Yes], [9, 1, 7, 7, Yes], [9, 2, 6, 6, Yes], [9, 3, 5, 5, Yes], [9, 4, 4, 4, Yes], [9, 5, 3, 3, Yes], [9, 6, 2, 2, Yes], [9, 7, 1, 1, Yes], [9, 8, 0, 0, Yes], [9, 9, 16, 16, Yes], [9, 10, 15, 15, Yes], [9, 11, 14, 14, Yes], [9, 12, 13, 13, Yes], [9, 13, 12, 12, Yes], [9, 14, 11, 11, Yes], [9, 15, 10, 10, Yes], [9, 16, 9, 9, Yes], [10, 0, 7, 7, Yes], [10, 1, 6, 6, Yes], [10, 2, 5, 5, Yes], [10, 3, 4, 4, Yes], [10, 4, 3, 3, Yes], [10, 5, 2, 2, Yes], [10, 6, 1, 1, Yes], [10, 7, 0, 0, Yes], [10, 8, 16, 16, Yes], [10, 9, 15, 15, Yes], [10, 10, 14, 14, Yes], [10, 11, 13, 13, Yes], [10, 12, 12, 12, Yes], [10, 13, 11, 11, Yes], [10, 14, 10, 10, Yes], [10, 15, 9, 9, Yes], [10, 16, 8, 8, Yes], [11, 0, 6, 6, Yes], [11, 1, 5, 5, Yes], [11, 2, 4, 4, Yes], [11, 3, 3, 3, Yes], [11, 4, 2, 2, Yes], [11, 5, 1, 1, Yes], [11, 6, 0, 0, Yes], [11, 7, 16, 16, Yes], [11, 8, 15, 15, Yes], [11, 9, 14, 14, Yes], [11, 10, 13, 13, Yes], [11, 11, 12, 12, Yes], [11, 12, 11, 11, Yes], [11, 13, 10, 10, Yes], [11, 14, 9, 9, Yes], [11, 15, 8, 8, Yes], [11, 16, 7, 7, Yes], [12, 0, 5, 5, Yes], [12, 1, 4, 4, Yes], [12, 2, 3, 3, Yes], [12, 3, 2, 2, Yes], [12, 4, 1, 1, Yes], [12, 5, 0, 0, Yes], [12, 6, 16, 16, Yes], [12, 7, 15, 15, Yes], [12, 8, 14, 14, Yes], [12, 9, 13, 13, Yes], [12, 10, 12, 12, Yes], [12, 11, 11, 11, Yes], [12, 12, 10, 10, Yes], [12, 13, 9, 9, Yes], [12, 14, 8, 8, Yes], [12, 15, 7, 7, Yes], [12, 16, 6, 6, Yes], [13, 0, 4, 4, Yes], [13, 1, 3, 3, Yes], [13, 2, 2, 2, Yes], [13, 3, 1, 1, Yes], [13, 4, 0, 0, Yes], [13, 5, 16, 16, Yes], [13, 6, 15, 15, Yes], [13, 7, 14, 14, Yes], [13, 8, 13, 13, Yes], [13, 9, 12, 12, Yes], [13, 10, 11, 11, Yes], [13, 11, 10, 10, Yes], [13, 12, 9, 9, Yes], [13, 13, 8, 8, Yes], [13, 14, 7, 7, Yes], [13, 15, 6, 6, Yes], [13, 16, 5, 5, Yes], [14, 0, 3, 3, Yes], [14, 1, 2, 2, Yes], [14, 2, 1, 1, Yes], [14, 3, 0, 0, Yes], [14, 4, 16, 16, Yes], [14, 5, 15, 15, Yes], [14, 6, 14, 14, Yes], [14, 7, 13, 13, Yes], [14, 8, 12, 12, Yes], [14, 9, 11, 11, Yes], [14, 10, 10, 10, Yes], [14, 11, 9, 9, Yes], [14, 12, 8, 8, Yes], [14, 13, 7, 7, Yes], [14, 14, 6, 6, Yes], [14, 15, 5, 5, Yes], [14, 16, 4, 4, Yes], [15, 0, 2, 2, Yes], [15, 1, 1, 1, Yes], [15, 2, 0, 0, Yes], [15, 3, 16, 16, Yes], [15, 4, 15, 15, Yes], [15, 5, 14, 14, Yes], [15, 6, 13, 13, Yes], [15, 7, 12, 12, Yes], [15, 8, 11, 11, Yes], [15, 9, 10, 10, Yes], [15, 10, 9, 9, Yes], [15, 11, 8, 8, Yes], [15, 12, 7, 7, Yes], [15, 13, 6, 6, Yes], [15, 14, 5, 5, Yes], [15, 15, 4, 4, Yes], [15, 16, 3, 3, Yes], [16, 0, 1, 1, Yes], [16, 1, 0, 0, Yes], [16, 2, 16, 16, Yes], [16, 3, 15, 15, Yes], [16, 4, 14, 14, Yes], [16, 5, 13, 13, Yes], [16, 6, 12, 12, Yes], [16, 7, 11, 11, Yes], [16, 8, 10, 10, Yes], [16, 9, 9, 9, Yes], [16, 10, 8, 8, Yes], [16, 11, 7, 7, Yes], [16, 12, 6, 6, Yes], [16, 13, 5, 5, Yes], [16, 14, 4, 4, Yes], [16, 15, 3, 3, Yes], [16, 16, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-homomorphism-qualification · candidate
16. Qualify Both Groups and Every Product Pair before Applying Kernel Theorems
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]], target identity=0, and total mapping=[0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).
View answer and explanation
(18, 18, [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 17, 17, Yes], [0, 2, 16, 16, Yes], [0, 3, 15, 15, Yes], [0, 4, 14, 14, Yes], [0, 5, 13, 13, Yes], [0, 6, 12, 12, Yes], [0, 7, 11, 11, Yes], [0, 8, 10, 10, Yes], [0, 9, 9, 9, Yes], [0, 10, 8, 8, Yes], [0, 11, 7, 7, Yes], [0, 12, 6, 6, Yes], [0, 13, 5, 5, Yes], [0, 14, 4, 4, Yes], [0, 15, 3, 3, Yes], [0, 16, 2, 2, Yes], [0, 17, 1, 1, Yes], [1, 0, 17, 17, Yes], [1, 1, 16, 16, Yes], [1, 2, 15, 15, Yes], [1, 3, 14, 14, Yes], [1, 4, 13, 13, Yes], [1, 5, 12, 12, Yes], [1, 6, 11, 11, Yes], [1, 7, 10, 10, Yes], [1, 8, 9, 9, Yes], [1, 9, 8, 8, Yes], [1, 10, 7, 7, Yes], [1, 11, 6, 6, Yes], [1, 12, 5, 5, Yes], [1, 13, 4, 4, Yes], [1, 14, 3, 3, Yes], [1, 15, 2, 2, Yes], [1, 16, 1, 1, Yes], [1, 17, 0, 0, Yes], [2, 0, 16, 16, Yes], [2, 1, 15, 15, Yes], [2, 2, 14, 14, Yes], [2, 3, 13, 13, Yes], [2, 4, 12, 12, Yes], [2, 5, 11, 11, Yes], [2, 6, 10, 10, Yes], [2, 7, 9, 9, Yes], [2, 8, 8, 8, Yes], [2, 9, 7, 7, Yes], [2, 10, 6, 6, Yes], [2, 11, 5, 5, Yes], [2, 12, 4, 4, Yes], [2, 13, 3, 3, Yes], [2, 14, 2, 2, Yes], [2, 15, 1, 1, Yes], [2, 16, 0, 0, Yes], [2, 17, 17, 17, Yes], [3, 0, 15, 15, Yes], [3, 1, 14, 14, Yes], [3, 2, 13, 13, Yes], [3, 3, 12, 12, Yes], [3, 4, 11, 11, Yes], [3, 5, 10, 10, Yes], [3, 6, 9, 9, Yes], [3, 7, 8, 8, Yes], [3, 8, 7, 7, Yes], [3, 9, 6, 6, Yes], [3, 10, 5, 5, Yes], [3, 11, 4, 4, Yes], [3, 12, 3, 3, Yes], [3, 13, 2, 2, Yes], [3, 14, 1, 1, Yes], [3, 15, 0, 0, Yes], [3, 16, 17, 17, Yes], [3, 17, 16, 16, Yes], [4, 0, 14, 14, Yes], [4, 1, 13, 13, Yes], [4, 2, 12, 12, Yes], [4, 3, 11, 11, Yes], [4, 4, 10, 10, Yes], [4, 5, 9, 9, Yes], [4, 6, 8, 8, Yes], [4, 7, 7, 7, Yes], [4, 8, 6, 6, Yes], [4, 9, 5, 5, Yes], [4, 10, 4, 4, Yes], [4, 11, 3, 3, Yes], [4, 12, 2, 2, Yes], [4, 13, 1, 1, Yes], [4, 14, 0, 0, Yes], [4, 15, 17, 17, Yes], [4, 16, 16, 16, Yes], [4, 17, 15, 15, Yes], [5, 0, 13, 13, Yes], [5, 1, 12, 12, Yes], [5, 2, 11, 11, Yes], [5, 3, 10, 10, Yes], [5, 4, 9, 9, Yes], [5, 5, 8, 8, Yes], [5, 6, 7, 7, Yes], [5, 7, 6, 6, Yes], [5, 8, 5, 5, Yes], [5, 9, 4, 4, Yes], [5, 10, 3, 3, Yes], [5, 11, 2, 2, Yes], [5, 12, 1, 1, Yes], [5, 13, 0, 0, Yes], [5, 14, 17, 17, Yes], [5, 15, 16, 16, Yes], [5, 16, 15, 15, Yes], [5, 17, 14, 14, Yes], [6, 0, 12, 12, Yes], [6, 1, 11, 11, Yes], [6, 2, 10, 10, Yes], [6, 3, 9, 9, Yes], [6, 4, 8, 8, Yes], [6, 5, 7, 7, Yes], [6, 6, 6, 6, Yes], [6, 7, 5, 5, Yes], [6, 8, 4, 4, Yes], [6, 9, 3, 3, Yes], [6, 10, 2, 2, Yes], [6, 11, 1, 1, Yes], [6, 12, 0, 0, Yes], [6, 13, 17, 17, Yes], [6, 14, 16, 16, Yes], [6, 15, 15, 15, Yes], [6, 16, 14, 14, Yes], [6, 17, 13, 13, Yes], [7, 0, 11, 11, Yes], [7, 1, 10, 10, Yes], [7, 2, 9, 9, Yes], [7, 3, 8, 8, Yes], [7, 4, 7, 7, Yes], [7, 5, 6, 6, Yes], [7, 6, 5, 5, Yes], [7, 7, 4, 4, Yes], [7, 8, 3, 3, Yes], [7, 9, 2, 2, Yes], [7, 10, 1, 1, Yes], [7, 11, 0, 0, Yes], [7, 12, 17, 17, Yes], [7, 13, 16, 16, Yes], [7, 14, 15, 15, Yes], [7, 15, 14, 14, Yes], [7, 16, 13, 13, Yes], [7, 17, 12, 12, Yes], [8, 0, 10, 10, Yes], [8, 1, 9, 9, Yes], [8, 2, 8, 8, Yes], [8, 3, 7, 7, Yes], [8, 4, 6, 6, Yes], [8, 5, 5, 5, Yes], [8, 6, 4, 4, Yes], [8, 7, 3, 3, Yes], [8, 8, 2, 2, Yes], [8, 9, 1, 1, Yes], [8, 10, 0, 0, Yes], [8, 11, 17, 17, Yes], [8, 12, 16, 16, Yes], [8, 13, 15, 15, Yes], [8, 14, 14, 14, Yes], [8, 15, 13, 13, Yes], [8, 16, 12, 12, Yes], [8, 17, 11, 11, Yes], [9, 0, 9, 9, Yes], [9, 1, 8, 8, Yes], [9, 2, 7, 7, Yes], [9, 3, 6, 6, Yes], [9, 4, 5, 5, Yes], [9, 5, 4, 4, Yes], [9, 6, 3, 3, Yes], [9, 7, 2, 2, Yes], [9, 8, 1, 1, Yes], [9, 9, 0, 0, Yes], [9, 10, 17, 17, Yes], [9, 11, 16, 16, Yes], [9, 12, 15, 15, Yes], [9, 13, 14, 14, Yes], [9, 14, 13, 13, Yes], [9, 15, 12, 12, Yes], [9, 16, 11, 11, Yes], [9, 17, 10, 10, Yes], [10, 0, 8, 8, Yes], [10, 1, 7, 7, Yes], [10, 2, 6, 6, Yes], [10, 3, 5, 5, Yes], [10, 4, 4, 4, Yes], [10, 5, 3, 3, Yes], [10, 6, 2, 2, Yes], [10, 7, 1, 1, Yes], [10, 8, 0, 0, Yes], [10, 9, 17, 17, Yes], [10, 10, 16, 16, Yes], [10, 11, 15, 15, Yes], [10, 12, 14, 14, Yes], [10, 13, 13, 13, Yes], [10, 14, 12, 12, Yes], [10, 15, 11, 11, Yes], [10, 16, 10, 10, Yes], [10, 17, 9, 9, Yes], [11, 0, 7, 7, Yes], [11, 1, 6, 6, Yes], [11, 2, 5, 5, Yes], [11, 3, 4, 4, Yes], [11, 4, 3, 3, Yes], [11, 5, 2, 2, Yes], [11, 6, 1, 1, Yes], [11, 7, 0, 0, Yes], [11, 8, 17, 17, Yes], [11, 9, 16, 16, Yes], [11, 10, 15, 15, Yes], [11, 11, 14, 14, Yes], [11, 12, 13, 13, Yes], [11, 13, 12, 12, Yes], [11, 14, 11, 11, Yes], [11, 15, 10, 10, Yes], [11, 16, 9, 9, Yes], [11, 17, 8, 8, Yes], [12, 0, 6, 6, Yes], [12, 1, 5, 5, Yes], [12, 2, 4, 4, Yes], [12, 3, 3, 3, Yes], [12, 4, 2, 2, Yes], [12, 5, 1, 1, Yes], [12, 6, 0, 0, Yes], [12, 7, 17, 17, Yes], [12, 8, 16, 16, Yes], [12, 9, 15, 15, Yes], [12, 10, 14, 14, Yes], [12, 11, 13, 13, Yes], [12, 12, 12, 12, Yes], [12, 13, 11, 11, Yes], [12, 14, 10, 10, Yes], [12, 15, 9, 9, Yes], [12, 16, 8, 8, Yes], [12, 17, 7, 7, Yes], [13, 0, 5, 5, Yes], [13, 1, 4, 4, Yes], [13, 2, 3, 3, Yes], [13, 3, 2, 2, Yes], [13, 4, 1, 1, Yes], [13, 5, 0, 0, Yes], [13, 6, 17, 17, Yes], [13, 7, 16, 16, Yes], [13, 8, 15, 15, Yes], [13, 9, 14, 14, Yes], [13, 10, 13, 13, Yes], [13, 11, 12, 12, Yes], [13, 12, 11, 11, Yes], [13, 13, 10, 10, Yes], [13, 14, 9, 9, Yes], [13, 15, 8, 8, Yes], [13, 16, 7, 7, Yes], [13, 17, 6, 6, Yes], [14, 0, 4, 4, Yes], [14, 1, 3, 3, Yes], [14, 2, 2, 2, Yes], [14, 3, 1, 1, Yes], [14, 4, 0, 0, Yes], [14, 5, 17, 17, Yes], [14, 6, 16, 16, Yes], [14, 7, 15, 15, Yes], [14, 8, 14, 14, Yes], [14, 9, 13, 13, Yes], [14, 10, 12, 12, Yes], [14, 11, 11, 11, Yes], [14, 12, 10, 10, Yes], [14, 13, 9, 9, Yes], [14, 14, 8, 8, Yes], [14, 15, 7, 7, Yes], [14, 16, 6, 6, Yes], [14, 17, 5, 5, Yes], [15, 0, 3, 3, Yes], [15, 1, 2, 2, Yes], [15, 2, 1, 1, Yes], [15, 3, 0, 0, Yes], [15, 4, 17, 17, Yes], [15, 5, 16, 16, Yes], [15, 6, 15, 15, Yes], [15, 7, 14, 14, Yes], [15, 8, 13, 13, Yes], [15, 9, 12, 12, Yes], [15, 10, 11, 11, Yes], [15, 11, 10, 10, Yes], [15, 12, 9, 9, Yes], [15, 13, 8, 8, Yes], [15, 14, 7, 7, Yes], [15, 15, 6, 6, Yes], [15, 16, 5, 5, Yes], [15, 17, 4, 4, Yes], [16, 0, 2, 2, Yes], [16, 1, 1, 1, Yes], [16, 2, 0, 0, Yes], [16, 3, 17, 17, Yes], [16, 4, 16, 16, Yes], [16, 5, 15, 15, Yes], [16, 6, 14, 14, Yes], [16, 7, 13, 13, Yes], [16, 8, 12, 12, Yes], [16, 9, 11, 11, Yes], [16, 10, 10, 10, Yes], [16, 11, 9, 9, Yes], [16, 12, 8, 8, Yes], [16, 13, 7, 7, Yes], [16, 14, 6, 6, Yes], [16, 15, 5, 5, Yes], [16, 16, 4, 4, Yes], [16, 17, 3, 3, Yes], [17, 0, 1, 1, Yes], [17, 1, 0, 0, Yes], [17, 2, 17, 17, Yes], [17, 3, 16, 16, Yes], [17, 4, 15, 15, Yes], [17, 5, 14, 14, Yes], [17, 6, 13, 13, Yes], [17, 7, 12, 12, Yes], [17, 8, 11, 11, Yes], [17, 9, 10, 10, Yes], [17, 10, 9, 9, Yes], [17, 11, 8, 8, Yes], [17, 12, 7, 7, Yes], [17, 13, 6, 6, Yes], [17, 14, 5, 5, Yes], [17, 15, 4, 4, Yes], [17, 16, 3, 3, Yes], [17, 17, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)
List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (18, 18, [0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 17, 17, Yes], [0, 2, 16, 16, Yes], [0, 3, 15, 15, Yes], [0, 4, 14, 14, Yes], [0, 5, 13, 13, Yes], [0, 6, 12, 12, Yes], [0, 7, 11, 11, Yes], [0, 8, 10, 10, Yes], [0, 9, 9, 9, Yes], [0, 10, 8, 8, Yes], [0, 11, 7, 7, Yes], [0, 12, 6, 6, Yes], [0, 13, 5, 5, Yes], [0, 14, 4, 4, Yes], [0, 15, 3, 3, Yes], [0, 16, 2, 2, Yes], [0, 17, 1, 1, Yes], [1, 0, 17, 17, Yes], [1, 1, 16, 16, Yes], [1, 2, 15, 15, Yes], [1, 3, 14, 14, Yes], [1, 4, 13, 13, Yes], [1, 5, 12, 12, Yes], [1, 6, 11, 11, Yes], [1, 7, 10, 10, Yes], [1, 8, 9, 9, Yes], [1, 9, 8, 8, Yes], [1, 10, 7, 7, Yes], [1, 11, 6, 6, Yes], [1, 12, 5, 5, Yes], [1, 13, 4, 4, Yes], [1, 14, 3, 3, Yes], [1, 15, 2, 2, Yes], [1, 16, 1, 1, Yes], [1, 17, 0, 0, Yes], [2, 0, 16, 16, Yes], [2, 1, 15, 15, Yes], [2, 2, 14, 14, Yes], [2, 3, 13, 13, Yes], [2, 4, 12, 12, Yes], [2, 5, 11, 11, Yes], [2, 6, 10, 10, Yes], [2, 7, 9, 9, Yes], [2, 8, 8, 8, Yes], [2, 9, 7, 7, Yes], [2, 10, 6, 6, Yes], [2, 11, 5, 5, Yes], [2, 12, 4, 4, Yes], [2, 13, 3, 3, Yes], [2, 14, 2, 2, Yes], [2, 15, 1, 1, Yes], [2, 16, 0, 0, Yes], [2, 17, 17, 17, Yes], [3, 0, 15, 15, Yes], [3, 1, 14, 14, Yes], [3, 2, 13, 13, Yes], [3, 3, 12, 12, Yes], [3, 4, 11, 11, Yes], [3, 5, 10, 10, Yes], [3, 6, 9, 9, Yes], [3, 7, 8, 8, Yes], [3, 8, 7, 7, Yes], [3, 9, 6, 6, Yes], [3, 10, 5, 5, Yes], [3, 11, 4, 4, Yes], [3, 12, 3, 3, Yes], [3, 13, 2, 2, Yes], [3, 14, 1, 1, Yes], [3, 15, 0, 0, Yes], [3, 16, 17, 17, Yes], [3, 17, 16, 16, Yes], [4, 0, 14, 14, Yes], [4, 1, 13, 13, Yes], [4, 2, 12, 12, Yes], [4, 3, 11, 11, Yes], [4, 4, 10, 10, Yes], [4, 5, 9, 9, Yes], [4, 6, 8, 8, Yes], [4, 7, 7, 7, Yes], [4, 8, 6, 6, Yes], [4, 9, 5, 5, Yes], [4, 10, 4, 4, Yes], [4, 11, 3, 3, Yes], [4, 12, 2, 2, Yes], [4, 13, 1, 1, Yes], [4, 14, 0, 0, Yes], [4, 15, 17, 17, Yes], [4, 16, 16, 16, Yes], [4, 17, 15, 15, Yes], [5, 0, 13, 13, Yes], [5, 1, 12, 12, Yes], [5, 2, 11, 11, Yes], [5, 3, 10, 10, Yes], [5, 4, 9, 9, Yes], [5, 5, 8, 8, Yes], [5, 6, 7, 7, Yes], [5, 7, 6, 6, Yes], [5, 8, 5, 5, Yes], [5, 9, 4, 4, Yes], [5, 10, 3, 3, Yes], [5, 11, 2, 2, Yes], [5, 12, 1, 1, Yes], [5, 13, 0, 0, Yes], [5, 14, 17, 17, Yes], [5, 15, 16, 16, Yes], [5, 16, 15, 15, Yes], [5, 17, 14, 14, Yes], [6, 0, 12, 12, Yes], [6, 1, 11, 11, Yes], [6, 2, 10, 10, Yes], [6, 3, 9, 9, Yes], [6, 4, 8, 8, Yes], [6, 5, 7, 7, Yes], [6, 6, 6, 6, Yes], [6, 7, 5, 5, Yes], [6, 8, 4, 4, Yes], [6, 9, 3, 3, Yes], [6, 10, 2, 2, Yes], [6, 11, 1, 1, Yes], [6, 12, 0, 0, Yes], [6, 13, 17, 17, Yes], [6, 14, 16, 16, Yes], [6, 15, 15, 15, Yes], [6, 16, 14, 14, Yes], [6, 17, 13, 13, Yes], [7, 0, 11, 11, Yes], [7, 1, 10, 10, Yes], [7, 2, 9, 9, Yes], [7, 3, 8, 8, Yes], [7, 4, 7, 7, Yes], [7, 5, 6, 6, Yes], [7, 6, 5, 5, Yes], [7, 7, 4, 4, Yes], [7, 8, 3, 3, Yes], [7, 9, 2, 2, Yes], [7, 10, 1, 1, Yes], [7, 11, 0, 0, Yes], [7, 12, 17, 17, Yes], [7, 13, 16, 16, Yes], [7, 14, 15, 15, Yes], [7, 15, 14, 14, Yes], [7, 16, 13, 13, Yes], [7, 17, 12, 12, Yes], [8, 0, 10, 10, Yes], [8, 1, 9, 9, Yes], [8, 2, 8, 8, Yes], [8, 3, 7, 7, Yes], [8, 4, 6, 6, Yes], [8, 5, 5, 5, Yes], [8, 6, 4, 4, Yes], [8, 7, 3, 3, Yes], [8, 8, 2, 2, Yes], [8, 9, 1, 1, Yes], [8, 10, 0, 0, Yes], [8, 11, 17, 17, Yes], [8, 12, 16, 16, Yes], [8, 13, 15, 15, Yes], [8, 14, 14, 14, Yes], [8, 15, 13, 13, Yes], [8, 16, 12, 12, Yes], [8, 17, 11, 11, Yes], [9, 0, 9, 9, Yes], [9, 1, 8, 8, Yes], [9, 2, 7, 7, Yes], [9, 3, 6, 6, Yes], [9, 4, 5, 5, Yes], [9, 5, 4, 4, Yes], [9, 6, 3, 3, Yes], [9, 7, 2, 2, Yes], [9, 8, 1, 1, Yes], [9, 9, 0, 0, Yes], [9, 10, 17, 17, Yes], [9, 11, 16, 16, Yes], [9, 12, 15, 15, Yes], [9, 13, 14, 14, Yes], [9, 14, 13, 13, Yes], [9, 15, 12, 12, Yes], [9, 16, 11, 11, Yes], [9, 17, 10, 10, Yes], [10, 0, 8, 8, Yes], [10, 1, 7, 7, Yes], [10, 2, 6, 6, Yes], [10, 3, 5, 5, Yes], [10, 4, 4, 4, Yes], [10, 5, 3, 3, Yes], [10, 6, 2, 2, Yes], [10, 7, 1, 1, Yes], [10, 8, 0, 0, Yes], [10, 9, 17, 17, Yes], [10, 10, 16, 16, Yes], [10, 11, 15, 15, Yes], [10, 12, 14, 14, Yes], [10, 13, 13, 13, Yes], [10, 14, 12, 12, Yes], [10, 15, 11, 11, Yes], [10, 16, 10, 10, Yes], [10, 17, 9, 9, Yes], [11, 0, 7, 7, Yes], [11, 1, 6, 6, Yes], [11, 2, 5, 5, Yes], [11, 3, 4, 4, Yes], [11, 4, 3, 3, Yes], [11, 5, 2, 2, Yes], [11, 6, 1, 1, Yes], [11, 7, 0, 0, Yes], [11, 8, 17, 17, Yes], [11, 9, 16, 16, Yes], [11, 10, 15, 15, Yes], [11, 11, 14, 14, Yes], [11, 12, 13, 13, Yes], [11, 13, 12, 12, Yes], [11, 14, 11, 11, Yes], [11, 15, 10, 10, Yes], [11, 16, 9, 9, Yes], [11, 17, 8, 8, Yes], [12, 0, 6, 6, Yes], [12, 1, 5, 5, Yes], [12, 2, 4, 4, Yes], [12, 3, 3, 3, Yes], [12, 4, 2, 2, Yes], [12, 5, 1, 1, Yes], [12, 6, 0, 0, Yes], [12, 7, 17, 17, Yes], [12, 8, 16, 16, Yes], [12, 9, 15, 15, Yes], [12, 10, 14, 14, Yes], [12, 11, 13, 13, Yes], [12, 12, 12, 12, Yes], [12, 13, 11, 11, Yes], [12, 14, 10, 10, Yes], [12, 15, 9, 9, Yes], [12, 16, 8, 8, Yes], [12, 17, 7, 7, Yes], [13, 0, 5, 5, Yes], [13, 1, 4, 4, Yes], [13, 2, 3, 3, Yes], [13, 3, 2, 2, Yes], [13, 4, 1, 1, Yes], [13, 5, 0, 0, Yes], [13, 6, 17, 17, Yes], [13, 7, 16, 16, Yes], [13, 8, 15, 15, Yes], [13, 9, 14, 14, Yes], [13, 10, 13, 13, Yes], [13, 11, 12, 12, Yes], [13, 12, 11, 11, Yes], [13, 13, 10, 10, Yes], [13, 14, 9, 9, Yes], [13, 15, 8, 8, Yes], [13, 16, 7, 7, Yes], [13, 17, 6, 6, Yes], [14, 0, 4, 4, Yes], [14, 1, 3, 3, Yes], [14, 2, 2, 2, Yes], [14, 3, 1, 1, Yes], [14, 4, 0, 0, Yes], [14, 5, 17, 17, Yes], [14, 6, 16, 16, Yes], [14, 7, 15, 15, Yes], [14, 8, 14, 14, Yes], [14, 9, 13, 13, Yes], [14, 10, 12, 12, Yes], [14, 11, 11, 11, Yes], [14, 12, 10, 10, Yes], [14, 13, 9, 9, Yes], [14, 14, 8, 8, Yes], [14, 15, 7, 7, Yes], [14, 16, 6, 6, Yes], [14, 17, 5, 5, Yes], [15, 0, 3, 3, Yes], [15, 1, 2, 2, Yes], [15, 2, 1, 1, Yes], [15, 3, 0, 0, Yes], [15, 4, 17, 17, Yes], [15, 5, 16, 16, Yes], [15, 6, 15, 15, Yes], [15, 7, 14, 14, Yes], [15, 8, 13, 13, Yes], [15, 9, 12, 12, Yes], [15, 10, 11, 11, Yes], [15, 11, 10, 10, Yes], [15, 12, 9, 9, Yes], [15, 13, 8, 8, Yes], [15, 14, 7, 7, Yes], [15, 15, 6, 6, Yes], [15, 16, 5, 5, Yes], [15, 17, 4, 4, Yes], [16, 0, 2, 2, Yes], [16, 1, 1, 1, Yes], [16, 2, 0, 0, Yes], [16, 3, 17, 17, Yes], [16, 4, 16, 16, Yes], [16, 5, 15, 15, Yes], [16, 6, 14, 14, Yes], [16, 7, 13, 13, Yes], [16, 8, 12, 12, Yes], [16, 9, 11, 11, Yes], [16, 10, 10, 10, Yes], [16, 11, 9, 9, Yes], [16, 12, 8, 8, Yes], [16, 13, 7, 7, Yes], [16, 14, 6, 6, Yes], [16, 15, 5, 5, Yes], [16, 16, 4, 4, Yes], [16, 17, 3, 3, Yes], [17, 0, 1, 1, Yes], [17, 1, 0, 0, Yes], [17, 2, 17, 17, Yes], [17, 3, 16, 16, Yes], [17, 4, 15, 15, Yes], [17, 5, 14, 14, Yes], [17, 6, 13, 13, Yes], [17, 7, 12, 12, Yes], [17, 8, 11, 11, Yes], [17, 9, 10, 10, Yes], [17, 10, 9, 9, Yes], [17, 11, 8, 8, Yes], [17, 12, 7, 7, Yes], [17, 13, 6, 6, Yes], [17, 14, 5, 5, Yes], [17, 15, 4, 4, Yes], [17, 16, 3, 3, Yes], [17, 17, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-homomorphism-qualification · candidate
17. Qualify Both Groups and Every Product Pair before Applying Kernel Theorems
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]], target identity=0, and total mapping=[0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).
View answer and explanation
(19, 19, [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 18, 18, Yes], [0, 2, 17, 17, Yes], [0, 3, 16, 16, Yes], [0, 4, 15, 15, Yes], [0, 5, 14, 14, Yes], [0, 6, 13, 13, Yes], [0, 7, 12, 12, Yes], [0, 8, 11, 11, Yes], [0, 9, 10, 10, Yes], [0, 10, 9, 9, Yes], [0, 11, 8, 8, Yes], [0, 12, 7, 7, Yes], [0, 13, 6, 6, Yes], [0, 14, 5, 5, Yes], [0, 15, 4, 4, Yes], [0, 16, 3, 3, Yes], [0, 17, 2, 2, Yes], [0, 18, 1, 1, Yes], [1, 0, 18, 18, Yes], [1, 1, 17, 17, Yes], [1, 2, 16, 16, Yes], [1, 3, 15, 15, Yes], [1, 4, 14, 14, Yes], [1, 5, 13, 13, Yes], [1, 6, 12, 12, Yes], [1, 7, 11, 11, Yes], [1, 8, 10, 10, Yes], [1, 9, 9, 9, Yes], [1, 10, 8, 8, Yes], [1, 11, 7, 7, Yes], [1, 12, 6, 6, Yes], [1, 13, 5, 5, Yes], [1, 14, 4, 4, Yes], [1, 15, 3, 3, Yes], [1, 16, 2, 2, Yes], [1, 17, 1, 1, Yes], [1, 18, 0, 0, Yes], [2, 0, 17, 17, Yes], [2, 1, 16, 16, Yes], [2, 2, 15, 15, Yes], [2, 3, 14, 14, Yes], [2, 4, 13, 13, Yes], [2, 5, 12, 12, Yes], [2, 6, 11, 11, Yes], [2, 7, 10, 10, Yes], [2, 8, 9, 9, Yes], [2, 9, 8, 8, Yes], [2, 10, 7, 7, Yes], [2, 11, 6, 6, Yes], [2, 12, 5, 5, Yes], [2, 13, 4, 4, Yes], [2, 14, 3, 3, Yes], [2, 15, 2, 2, Yes], [2, 16, 1, 1, Yes], [2, 17, 0, 0, Yes], [2, 18, 18, 18, Yes], [3, 0, 16, 16, Yes], [3, 1, 15, 15, Yes], [3, 2, 14, 14, Yes], [3, 3, 13, 13, Yes], [3, 4, 12, 12, Yes], [3, 5, 11, 11, Yes], [3, 6, 10, 10, Yes], [3, 7, 9, 9, Yes], [3, 8, 8, 8, Yes], [3, 9, 7, 7, Yes], [3, 10, 6, 6, Yes], [3, 11, 5, 5, Yes], [3, 12, 4, 4, Yes], [3, 13, 3, 3, Yes], [3, 14, 2, 2, Yes], [3, 15, 1, 1, Yes], [3, 16, 0, 0, Yes], [3, 17, 18, 18, Yes], [3, 18, 17, 17, Yes], [4, 0, 15, 15, Yes], [4, 1, 14, 14, Yes], [4, 2, 13, 13, Yes], [4, 3, 12, 12, Yes], [4, 4, 11, 11, Yes], [4, 5, 10, 10, Yes], [4, 6, 9, 9, Yes], [4, 7, 8, 8, Yes], [4, 8, 7, 7, Yes], [4, 9, 6, 6, Yes], [4, 10, 5, 5, Yes], [4, 11, 4, 4, Yes], [4, 12, 3, 3, Yes], [4, 13, 2, 2, Yes], [4, 14, 1, 1, Yes], [4, 15, 0, 0, Yes], [4, 16, 18, 18, Yes], [4, 17, 17, 17, Yes], [4, 18, 16, 16, Yes], [5, 0, 14, 14, Yes], [5, 1, 13, 13, Yes], [5, 2, 12, 12, Yes], [5, 3, 11, 11, Yes], [5, 4, 10, 10, Yes], [5, 5, 9, 9, Yes], [5, 6, 8, 8, Yes], [5, 7, 7, 7, Yes], [5, 8, 6, 6, Yes], [5, 9, 5, 5, Yes], [5, 10, 4, 4, Yes], [5, 11, 3, 3, Yes], [5, 12, 2, 2, Yes], [5, 13, 1, 1, Yes], [5, 14, 0, 0, Yes], [5, 15, 18, 18, Yes], [5, 16, 17, 17, Yes], [5, 17, 16, 16, Yes], [5, 18, 15, 15, Yes], [6, 0, 13, 13, Yes], [6, 1, 12, 12, Yes], [6, 2, 11, 11, Yes], [6, 3, 10, 10, Yes], [6, 4, 9, 9, Yes], [6, 5, 8, 8, Yes], [6, 6, 7, 7, Yes], [6, 7, 6, 6, Yes], [6, 8, 5, 5, Yes], [6, 9, 4, 4, Yes], [6, 10, 3, 3, Yes], [6, 11, 2, 2, Yes], [6, 12, 1, 1, Yes], [6, 13, 0, 0, Yes], [6, 14, 18, 18, Yes], [6, 15, 17, 17, Yes], [6, 16, 16, 16, Yes], [6, 17, 15, 15, Yes], [6, 18, 14, 14, Yes], [7, 0, 12, 12, Yes], [7, 1, 11, 11, Yes], [7, 2, 10, 10, Yes], [7, 3, 9, 9, Yes], [7, 4, 8, 8, Yes], [7, 5, 7, 7, Yes], [7, 6, 6, 6, Yes], [7, 7, 5, 5, Yes], [7, 8, 4, 4, Yes], [7, 9, 3, 3, Yes], [7, 10, 2, 2, Yes], [7, 11, 1, 1, Yes], [7, 12, 0, 0, Yes], [7, 13, 18, 18, Yes], [7, 14, 17, 17, Yes], [7, 15, 16, 16, Yes], [7, 16, 15, 15, Yes], [7, 17, 14, 14, Yes], [7, 18, 13, 13, Yes], [8, 0, 11, 11, Yes], [8, 1, 10, 10, Yes], [8, 2, 9, 9, Yes], [8, 3, 8, 8, Yes], [8, 4, 7, 7, Yes], [8, 5, 6, 6, Yes], [8, 6, 5, 5, Yes], [8, 7, 4, 4, Yes], [8, 8, 3, 3, Yes], [8, 9, 2, 2, Yes], [8, 10, 1, 1, Yes], [8, 11, 0, 0, Yes], [8, 12, 18, 18, Yes], [8, 13, 17, 17, Yes], [8, 14, 16, 16, Yes], [8, 15, 15, 15, Yes], [8, 16, 14, 14, Yes], [8, 17, 13, 13, Yes], [8, 18, 12, 12, Yes], [9, 0, 10, 10, Yes], [9, 1, 9, 9, Yes], [9, 2, 8, 8, Yes], [9, 3, 7, 7, Yes], [9, 4, 6, 6, Yes], [9, 5, 5, 5, Yes], [9, 6, 4, 4, Yes], [9, 7, 3, 3, Yes], [9, 8, 2, 2, Yes], [9, 9, 1, 1, Yes], [9, 10, 0, 0, Yes], [9, 11, 18, 18, Yes], [9, 12, 17, 17, Yes], [9, 13, 16, 16, Yes], [9, 14, 15, 15, Yes], [9, 15, 14, 14, Yes], [9, 16, 13, 13, Yes], [9, 17, 12, 12, Yes], [9, 18, 11, 11, Yes], [10, 0, 9, 9, Yes], [10, 1, 8, 8, Yes], [10, 2, 7, 7, Yes], [10, 3, 6, 6, Yes], [10, 4, 5, 5, Yes], [10, 5, 4, 4, Yes], [10, 6, 3, 3, Yes], [10, 7, 2, 2, Yes], [10, 8, 1, 1, Yes], [10, 9, 0, 0, Yes], [10, 10, 18, 18, Yes], [10, 11, 17, 17, Yes], [10, 12, 16, 16, Yes], [10, 13, 15, 15, Yes], [10, 14, 14, 14, Yes], [10, 15, 13, 13, Yes], [10, 16, 12, 12, Yes], [10, 17, 11, 11, Yes], [10, 18, 10, 10, Yes], [11, 0, 8, 8, Yes], [11, 1, 7, 7, Yes], [11, 2, 6, 6, Yes], [11, 3, 5, 5, Yes], [11, 4, 4, 4, Yes], [11, 5, 3, 3, Yes], [11, 6, 2, 2, Yes], [11, 7, 1, 1, Yes], [11, 8, 0, 0, Yes], [11, 9, 18, 18, Yes], [11, 10, 17, 17, Yes], [11, 11, 16, 16, Yes], [11, 12, 15, 15, Yes], [11, 13, 14, 14, Yes], [11, 14, 13, 13, Yes], [11, 15, 12, 12, Yes], [11, 16, 11, 11, Yes], [11, 17, 10, 10, Yes], [11, 18, 9, 9, Yes], [12, 0, 7, 7, Yes], [12, 1, 6, 6, Yes], [12, 2, 5, 5, Yes], [12, 3, 4, 4, Yes], [12, 4, 3, 3, Yes], [12, 5, 2, 2, Yes], [12, 6, 1, 1, Yes], [12, 7, 0, 0, Yes], [12, 8, 18, 18, Yes], [12, 9, 17, 17, Yes], [12, 10, 16, 16, Yes], [12, 11, 15, 15, Yes], [12, 12, 14, 14, Yes], [12, 13, 13, 13, Yes], [12, 14, 12, 12, Yes], [12, 15, 11, 11, Yes], [12, 16, 10, 10, Yes], [12, 17, 9, 9, Yes], [12, 18, 8, 8, Yes], [13, 0, 6, 6, Yes], [13, 1, 5, 5, Yes], [13, 2, 4, 4, Yes], [13, 3, 3, 3, Yes], [13, 4, 2, 2, Yes], [13, 5, 1, 1, Yes], [13, 6, 0, 0, Yes], [13, 7, 18, 18, Yes], [13, 8, 17, 17, Yes], [13, 9, 16, 16, Yes], [13, 10, 15, 15, Yes], [13, 11, 14, 14, Yes], [13, 12, 13, 13, Yes], [13, 13, 12, 12, Yes], [13, 14, 11, 11, Yes], [13, 15, 10, 10, Yes], [13, 16, 9, 9, Yes], [13, 17, 8, 8, Yes], [13, 18, 7, 7, Yes], [14, 0, 5, 5, Yes], [14, 1, 4, 4, Yes], [14, 2, 3, 3, Yes], [14, 3, 2, 2, Yes], [14, 4, 1, 1, Yes], [14, 5, 0, 0, Yes], [14, 6, 18, 18, Yes], [14, 7, 17, 17, Yes], [14, 8, 16, 16, Yes], [14, 9, 15, 15, Yes], [14, 10, 14, 14, Yes], [14, 11, 13, 13, Yes], [14, 12, 12, 12, Yes], [14, 13, 11, 11, Yes], [14, 14, 10, 10, Yes], [14, 15, 9, 9, Yes], [14, 16, 8, 8, Yes], [14, 17, 7, 7, Yes], [14, 18, 6, 6, Yes], [15, 0, 4, 4, Yes], [15, 1, 3, 3, Yes], [15, 2, 2, 2, Yes], [15, 3, 1, 1, Yes], [15, 4, 0, 0, Yes], [15, 5, 18, 18, Yes], [15, 6, 17, 17, Yes], [15, 7, 16, 16, Yes], [15, 8, 15, 15, Yes], [15, 9, 14, 14, Yes], [15, 10, 13, 13, Yes], [15, 11, 12, 12, Yes], [15, 12, 11, 11, Yes], [15, 13, 10, 10, Yes], [15, 14, 9, 9, Yes], [15, 15, 8, 8, Yes], [15, 16, 7, 7, Yes], [15, 17, 6, 6, Yes], [15, 18, 5, 5, Yes], [16, 0, 3, 3, Yes], [16, 1, 2, 2, Yes], [16, 2, 1, 1, Yes], [16, 3, 0, 0, Yes], [16, 4, 18, 18, Yes], [16, 5, 17, 17, Yes], [16, 6, 16, 16, Yes], [16, 7, 15, 15, Yes], [16, 8, 14, 14, Yes], [16, 9, 13, 13, Yes], [16, 10, 12, 12, Yes], [16, 11, 11, 11, Yes], [16, 12, 10, 10, Yes], [16, 13, 9, 9, Yes], [16, 14, 8, 8, Yes], [16, 15, 7, 7, Yes], [16, 16, 6, 6, Yes], [16, 17, 5, 5, Yes], [16, 18, 4, 4, Yes], [17, 0, 2, 2, Yes], [17, 1, 1, 1, Yes], [17, 2, 0, 0, Yes], [17, 3, 18, 18, Yes], [17, 4, 17, 17, Yes], [17, 5, 16, 16, Yes], [17, 6, 15, 15, Yes], [17, 7, 14, 14, Yes], [17, 8, 13, 13, Yes], [17, 9, 12, 12, Yes], [17, 10, 11, 11, Yes], [17, 11, 10, 10, Yes], [17, 12, 9, 9, Yes], [17, 13, 8, 8, Yes], [17, 14, 7, 7, Yes], [17, 15, 6, 6, Yes], [17, 16, 5, 5, Yes], [17, 17, 4, 4, Yes], [17, 18, 3, 3, Yes], [18, 0, 1, 1, Yes], [18, 1, 0, 0, Yes], [18, 2, 18, 18, Yes], [18, 3, 17, 17, Yes], [18, 4, 16, 16, Yes], [18, 5, 15, 15, Yes], [18, 6, 14, 14, Yes], [18, 7, 13, 13, Yes], [18, 8, 12, 12, Yes], [18, 9, 11, 11, Yes], [18, 10, 10, 10, Yes], [18, 11, 9, 9, Yes], [18, 12, 8, 8, Yes], [18, 13, 7, 7, Yes], [18, 14, 6, 6, Yes], [18, 15, 5, 5, Yes], [18, 16, 4, 4, Yes], [18, 17, 3, 3, Yes], [18, 18, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)
List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (19, 19, [0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 18, 18, Yes], [0, 2, 17, 17, Yes], [0, 3, 16, 16, Yes], [0, 4, 15, 15, Yes], [0, 5, 14, 14, Yes], [0, 6, 13, 13, Yes], [0, 7, 12, 12, Yes], [0, 8, 11, 11, Yes], [0, 9, 10, 10, Yes], [0, 10, 9, 9, Yes], [0, 11, 8, 8, Yes], [0, 12, 7, 7, Yes], [0, 13, 6, 6, Yes], [0, 14, 5, 5, Yes], [0, 15, 4, 4, Yes], [0, 16, 3, 3, Yes], [0, 17, 2, 2, Yes], [0, 18, 1, 1, Yes], [1, 0, 18, 18, Yes], [1, 1, 17, 17, Yes], [1, 2, 16, 16, Yes], [1, 3, 15, 15, Yes], [1, 4, 14, 14, Yes], [1, 5, 13, 13, Yes], [1, 6, 12, 12, Yes], [1, 7, 11, 11, Yes], [1, 8, 10, 10, Yes], [1, 9, 9, 9, Yes], [1, 10, 8, 8, Yes], [1, 11, 7, 7, Yes], [1, 12, 6, 6, Yes], [1, 13, 5, 5, Yes], [1, 14, 4, 4, Yes], [1, 15, 3, 3, Yes], [1, 16, 2, 2, Yes], [1, 17, 1, 1, Yes], [1, 18, 0, 0, Yes], [2, 0, 17, 17, Yes], [2, 1, 16, 16, Yes], [2, 2, 15, 15, Yes], [2, 3, 14, 14, Yes], [2, 4, 13, 13, Yes], [2, 5, 12, 12, Yes], [2, 6, 11, 11, Yes], [2, 7, 10, 10, Yes], [2, 8, 9, 9, Yes], [2, 9, 8, 8, Yes], [2, 10, 7, 7, Yes], [2, 11, 6, 6, Yes], [2, 12, 5, 5, Yes], [2, 13, 4, 4, Yes], [2, 14, 3, 3, Yes], [2, 15, 2, 2, Yes], [2, 16, 1, 1, Yes], [2, 17, 0, 0, Yes], [2, 18, 18, 18, Yes], [3, 0, 16, 16, Yes], [3, 1, 15, 15, Yes], [3, 2, 14, 14, Yes], [3, 3, 13, 13, Yes], [3, 4, 12, 12, Yes], [3, 5, 11, 11, Yes], [3, 6, 10, 10, Yes], [3, 7, 9, 9, Yes], [3, 8, 8, 8, Yes], [3, 9, 7, 7, Yes], [3, 10, 6, 6, Yes], [3, 11, 5, 5, Yes], [3, 12, 4, 4, Yes], [3, 13, 3, 3, Yes], [3, 14, 2, 2, Yes], [3, 15, 1, 1, Yes], [3, 16, 0, 0, Yes], [3, 17, 18, 18, Yes], [3, 18, 17, 17, Yes], [4, 0, 15, 15, Yes], [4, 1, 14, 14, Yes], [4, 2, 13, 13, Yes], [4, 3, 12, 12, Yes], [4, 4, 11, 11, Yes], [4, 5, 10, 10, Yes], [4, 6, 9, 9, Yes], [4, 7, 8, 8, Yes], [4, 8, 7, 7, Yes], [4, 9, 6, 6, Yes], [4, 10, 5, 5, Yes], [4, 11, 4, 4, Yes], [4, 12, 3, 3, Yes], [4, 13, 2, 2, Yes], [4, 14, 1, 1, Yes], [4, 15, 0, 0, Yes], [4, 16, 18, 18, Yes], [4, 17, 17, 17, Yes], [4, 18, 16, 16, Yes], [5, 0, 14, 14, Yes], [5, 1, 13, 13, Yes], [5, 2, 12, 12, Yes], [5, 3, 11, 11, Yes], [5, 4, 10, 10, Yes], [5, 5, 9, 9, Yes], [5, 6, 8, 8, Yes], [5, 7, 7, 7, Yes], [5, 8, 6, 6, Yes], [5, 9, 5, 5, Yes], [5, 10, 4, 4, Yes], [5, 11, 3, 3, Yes], [5, 12, 2, 2, Yes], [5, 13, 1, 1, Yes], [5, 14, 0, 0, Yes], [5, 15, 18, 18, Yes], [5, 16, 17, 17, Yes], [5, 17, 16, 16, Yes], [5, 18, 15, 15, Yes], [6, 0, 13, 13, Yes], [6, 1, 12, 12, Yes], [6, 2, 11, 11, Yes], [6, 3, 10, 10, Yes], [6, 4, 9, 9, Yes], [6, 5, 8, 8, Yes], [6, 6, 7, 7, Yes], [6, 7, 6, 6, Yes], [6, 8, 5, 5, Yes], [6, 9, 4, 4, Yes], [6, 10, 3, 3, Yes], [6, 11, 2, 2, Yes], [6, 12, 1, 1, Yes], [6, 13, 0, 0, Yes], [6, 14, 18, 18, Yes], [6, 15, 17, 17, Yes], [6, 16, 16, 16, Yes], [6, 17, 15, 15, Yes], [6, 18, 14, 14, Yes], [7, 0, 12, 12, Yes], [7, 1, 11, 11, Yes], [7, 2, 10, 10, Yes], [7, 3, 9, 9, Yes], [7, 4, 8, 8, Yes], [7, 5, 7, 7, Yes], [7, 6, 6, 6, Yes], [7, 7, 5, 5, Yes], [7, 8, 4, 4, Yes], [7, 9, 3, 3, Yes], [7, 10, 2, 2, Yes], [7, 11, 1, 1, Yes], [7, 12, 0, 0, Yes], [7, 13, 18, 18, Yes], [7, 14, 17, 17, Yes], [7, 15, 16, 16, Yes], [7, 16, 15, 15, Yes], [7, 17, 14, 14, Yes], [7, 18, 13, 13, Yes], [8, 0, 11, 11, Yes], [8, 1, 10, 10, Yes], [8, 2, 9, 9, Yes], [8, 3, 8, 8, Yes], [8, 4, 7, 7, Yes], [8, 5, 6, 6, Yes], [8, 6, 5, 5, Yes], [8, 7, 4, 4, Yes], [8, 8, 3, 3, Yes], [8, 9, 2, 2, Yes], [8, 10, 1, 1, Yes], [8, 11, 0, 0, Yes], [8, 12, 18, 18, Yes], [8, 13, 17, 17, Yes], [8, 14, 16, 16, Yes], [8, 15, 15, 15, Yes], [8, 16, 14, 14, Yes], [8, 17, 13, 13, Yes], [8, 18, 12, 12, Yes], [9, 0, 10, 10, Yes], [9, 1, 9, 9, Yes], [9, 2, 8, 8, Yes], [9, 3, 7, 7, Yes], [9, 4, 6, 6, Yes], [9, 5, 5, 5, Yes], [9, 6, 4, 4, Yes], [9, 7, 3, 3, Yes], [9, 8, 2, 2, Yes], [9, 9, 1, 1, Yes], [9, 10, 0, 0, Yes], [9, 11, 18, 18, Yes], [9, 12, 17, 17, Yes], [9, 13, 16, 16, Yes], [9, 14, 15, 15, Yes], [9, 15, 14, 14, Yes], [9, 16, 13, 13, Yes], [9, 17, 12, 12, Yes], [9, 18, 11, 11, Yes], [10, 0, 9, 9, Yes], [10, 1, 8, 8, Yes], [10, 2, 7, 7, Yes], [10, 3, 6, 6, Yes], [10, 4, 5, 5, Yes], [10, 5, 4, 4, Yes], [10, 6, 3, 3, Yes], [10, 7, 2, 2, Yes], [10, 8, 1, 1, Yes], [10, 9, 0, 0, Yes], [10, 10, 18, 18, Yes], [10, 11, 17, 17, Yes], [10, 12, 16, 16, Yes], [10, 13, 15, 15, Yes], [10, 14, 14, 14, Yes], [10, 15, 13, 13, Yes], [10, 16, 12, 12, Yes], [10, 17, 11, 11, Yes], [10, 18, 10, 10, Yes], [11, 0, 8, 8, Yes], [11, 1, 7, 7, Yes], [11, 2, 6, 6, Yes], [11, 3, 5, 5, Yes], [11, 4, 4, 4, Yes], [11, 5, 3, 3, Yes], [11, 6, 2, 2, Yes], [11, 7, 1, 1, Yes], [11, 8, 0, 0, Yes], [11, 9, 18, 18, Yes], [11, 10, 17, 17, Yes], [11, 11, 16, 16, Yes], [11, 12, 15, 15, Yes], [11, 13, 14, 14, Yes], [11, 14, 13, 13, Yes], [11, 15, 12, 12, Yes], [11, 16, 11, 11, Yes], [11, 17, 10, 10, Yes], [11, 18, 9, 9, Yes], [12, 0, 7, 7, Yes], [12, 1, 6, 6, Yes], [12, 2, 5, 5, Yes], [12, 3, 4, 4, Yes], [12, 4, 3, 3, Yes], [12, 5, 2, 2, Yes], [12, 6, 1, 1, Yes], [12, 7, 0, 0, Yes], [12, 8, 18, 18, Yes], [12, 9, 17, 17, Yes], [12, 10, 16, 16, Yes], [12, 11, 15, 15, Yes], [12, 12, 14, 14, Yes], [12, 13, 13, 13, Yes], [12, 14, 12, 12, Yes], [12, 15, 11, 11, Yes], [12, 16, 10, 10, Yes], [12, 17, 9, 9, Yes], [12, 18, 8, 8, Yes], [13, 0, 6, 6, Yes], [13, 1, 5, 5, Yes], [13, 2, 4, 4, Yes], [13, 3, 3, 3, Yes], [13, 4, 2, 2, Yes], [13, 5, 1, 1, Yes], [13, 6, 0, 0, Yes], [13, 7, 18, 18, Yes], [13, 8, 17, 17, Yes], [13, 9, 16, 16, Yes], [13, 10, 15, 15, Yes], [13, 11, 14, 14, Yes], [13, 12, 13, 13, Yes], [13, 13, 12, 12, Yes], [13, 14, 11, 11, Yes], [13, 15, 10, 10, Yes], [13, 16, 9, 9, Yes], [13, 17, 8, 8, Yes], [13, 18, 7, 7, Yes], [14, 0, 5, 5, Yes], [14, 1, 4, 4, Yes], [14, 2, 3, 3, Yes], [14, 3, 2, 2, Yes], [14, 4, 1, 1, Yes], [14, 5, 0, 0, Yes], [14, 6, 18, 18, Yes], [14, 7, 17, 17, Yes], [14, 8, 16, 16, Yes], [14, 9, 15, 15, Yes], [14, 10, 14, 14, Yes], [14, 11, 13, 13, Yes], [14, 12, 12, 12, Yes], [14, 13, 11, 11, Yes], [14, 14, 10, 10, Yes], [14, 15, 9, 9, Yes], [14, 16, 8, 8, Yes], [14, 17, 7, 7, Yes], [14, 18, 6, 6, Yes], [15, 0, 4, 4, Yes], [15, 1, 3, 3, Yes], [15, 2, 2, 2, Yes], [15, 3, 1, 1, Yes], [15, 4, 0, 0, Yes], [15, 5, 18, 18, Yes], [15, 6, 17, 17, Yes], [15, 7, 16, 16, Yes], [15, 8, 15, 15, Yes], [15, 9, 14, 14, Yes], [15, 10, 13, 13, Yes], [15, 11, 12, 12, Yes], [15, 12, 11, 11, Yes], [15, 13, 10, 10, Yes], [15, 14, 9, 9, Yes], [15, 15, 8, 8, Yes], [15, 16, 7, 7, Yes], [15, 17, 6, 6, Yes], [15, 18, 5, 5, Yes], [16, 0, 3, 3, Yes], [16, 1, 2, 2, Yes], [16, 2, 1, 1, Yes], [16, 3, 0, 0, Yes], [16, 4, 18, 18, Yes], [16, 5, 17, 17, Yes], [16, 6, 16, 16, Yes], [16, 7, 15, 15, Yes], [16, 8, 14, 14, Yes], [16, 9, 13, 13, Yes], [16, 10, 12, 12, Yes], [16, 11, 11, 11, Yes], [16, 12, 10, 10, Yes], [16, 13, 9, 9, Yes], [16, 14, 8, 8, Yes], [16, 15, 7, 7, Yes], [16, 16, 6, 6, Yes], [16, 17, 5, 5, Yes], [16, 18, 4, 4, Yes], [17, 0, 2, 2, Yes], [17, 1, 1, 1, Yes], [17, 2, 0, 0, Yes], [17, 3, 18, 18, Yes], [17, 4, 17, 17, Yes], [17, 5, 16, 16, Yes], [17, 6, 15, 15, Yes], [17, 7, 14, 14, Yes], [17, 8, 13, 13, Yes], [17, 9, 12, 12, Yes], [17, 10, 11, 11, Yes], [17, 11, 10, 10, Yes], [17, 12, 9, 9, Yes], [17, 13, 8, 8, Yes], [17, 14, 7, 7, Yes], [17, 15, 6, 6, Yes], [17, 16, 5, 5, Yes], [17, 17, 4, 4, Yes], [17, 18, 3, 3, Yes], [18, 0, 1, 1, Yes], [18, 1, 0, 0, Yes], [18, 2, 18, 18, Yes], [18, 3, 17, 17, Yes], [18, 4, 16, 16, Yes], [18, 5, 15, 15, Yes], [18, 6, 14, 14, Yes], [18, 7, 13, 13, Yes], [18, 8, 12, 12, Yes], [18, 9, 11, 11, Yes], [18, 10, 10, 10, Yes], [18, 11, 9, 9, Yes], [18, 12, 8, 8, Yes], [18, 13, 7, 7, Yes], [18, 14, 6, 6, Yes], [18, 15, 5, 5, Yes], [18, 16, 4, 4, Yes], [18, 17, 3, 3, Yes], [18, 18, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-homomorphism-qualification · candidate
18. Qualify Both Groups and Every Product Pair before Applying Kernel Theorems
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]], target identity=0, and total mapping=[0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).
View answer and explanation
(20, 20, [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 19, 19, Yes], [0, 2, 18, 18, Yes], [0, 3, 17, 17, Yes], [0, 4, 16, 16, Yes], [0, 5, 15, 15, Yes], [0, 6, 14, 14, Yes], [0, 7, 13, 13, Yes], [0, 8, 12, 12, Yes], [0, 9, 11, 11, Yes], [0, 10, 10, 10, Yes], [0, 11, 9, 9, Yes], [0, 12, 8, 8, Yes], [0, 13, 7, 7, Yes], [0, 14, 6, 6, Yes], [0, 15, 5, 5, Yes], [0, 16, 4, 4, Yes], [0, 17, 3, 3, Yes], [0, 18, 2, 2, Yes], [0, 19, 1, 1, Yes], [1, 0, 19, 19, Yes], [1, 1, 18, 18, Yes], [1, 2, 17, 17, Yes], [1, 3, 16, 16, Yes], [1, 4, 15, 15, Yes], [1, 5, 14, 14, Yes], [1, 6, 13, 13, Yes], [1, 7, 12, 12, Yes], [1, 8, 11, 11, Yes], [1, 9, 10, 10, Yes], [1, 10, 9, 9, Yes], [1, 11, 8, 8, Yes], [1, 12, 7, 7, Yes], [1, 13, 6, 6, Yes], [1, 14, 5, 5, Yes], [1, 15, 4, 4, Yes], [1, 16, 3, 3, Yes], [1, 17, 2, 2, Yes], [1, 18, 1, 1, Yes], [1, 19, 0, 0, Yes], [2, 0, 18, 18, Yes], [2, 1, 17, 17, Yes], [2, 2, 16, 16, Yes], [2, 3, 15, 15, Yes], [2, 4, 14, 14, Yes], [2, 5, 13, 13, Yes], [2, 6, 12, 12, Yes], [2, 7, 11, 11, Yes], [2, 8, 10, 10, Yes], [2, 9, 9, 9, Yes], [2, 10, 8, 8, Yes], [2, 11, 7, 7, Yes], [2, 12, 6, 6, Yes], [2, 13, 5, 5, Yes], [2, 14, 4, 4, Yes], [2, 15, 3, 3, Yes], [2, 16, 2, 2, Yes], [2, 17, 1, 1, Yes], [2, 18, 0, 0, Yes], [2, 19, 19, 19, Yes], [3, 0, 17, 17, Yes], [3, 1, 16, 16, Yes], [3, 2, 15, 15, Yes], [3, 3, 14, 14, Yes], [3, 4, 13, 13, Yes], [3, 5, 12, 12, Yes], [3, 6, 11, 11, Yes], [3, 7, 10, 10, Yes], [3, 8, 9, 9, Yes], [3, 9, 8, 8, Yes], [3, 10, 7, 7, Yes], [3, 11, 6, 6, Yes], [3, 12, 5, 5, Yes], [3, 13, 4, 4, Yes], [3, 14, 3, 3, Yes], [3, 15, 2, 2, Yes], [3, 16, 1, 1, Yes], [3, 17, 0, 0, Yes], [3, 18, 19, 19, Yes], [3, 19, 18, 18, Yes], [4, 0, 16, 16, Yes], [4, 1, 15, 15, Yes], [4, 2, 14, 14, Yes], [4, 3, 13, 13, Yes], [4, 4, 12, 12, Yes], [4, 5, 11, 11, Yes], [4, 6, 10, 10, Yes], [4, 7, 9, 9, Yes], [4, 8, 8, 8, Yes], [4, 9, 7, 7, Yes], [4, 10, 6, 6, Yes], [4, 11, 5, 5, Yes], [4, 12, 4, 4, Yes], [4, 13, 3, 3, Yes], [4, 14, 2, 2, Yes], [4, 15, 1, 1, Yes], [4, 16, 0, 0, Yes], [4, 17, 19, 19, Yes], [4, 18, 18, 18, Yes], [4, 19, 17, 17, Yes], [5, 0, 15, 15, Yes], [5, 1, 14, 14, Yes], [5, 2, 13, 13, Yes], [5, 3, 12, 12, Yes], [5, 4, 11, 11, Yes], [5, 5, 10, 10, Yes], [5, 6, 9, 9, Yes], [5, 7, 8, 8, Yes], [5, 8, 7, 7, Yes], [5, 9, 6, 6, Yes], [5, 10, 5, 5, Yes], [5, 11, 4, 4, Yes], [5, 12, 3, 3, Yes], [5, 13, 2, 2, Yes], [5, 14, 1, 1, Yes], [5, 15, 0, 0, Yes], [5, 16, 19, 19, Yes], [5, 17, 18, 18, Yes], [5, 18, 17, 17, Yes], [5, 19, 16, 16, Yes], [6, 0, 14, 14, Yes], [6, 1, 13, 13, Yes], [6, 2, 12, 12, Yes], [6, 3, 11, 11, Yes], [6, 4, 10, 10, Yes], [6, 5, 9, 9, Yes], [6, 6, 8, 8, Yes], [6, 7, 7, 7, Yes], [6, 8, 6, 6, Yes], [6, 9, 5, 5, Yes], [6, 10, 4, 4, Yes], [6, 11, 3, 3, Yes], [6, 12, 2, 2, Yes], [6, 13, 1, 1, Yes], [6, 14, 0, 0, Yes], [6, 15, 19, 19, Yes], [6, 16, 18, 18, Yes], [6, 17, 17, 17, Yes], [6, 18, 16, 16, Yes], [6, 19, 15, 15, Yes], [7, 0, 13, 13, Yes], [7, 1, 12, 12, Yes], [7, 2, 11, 11, Yes], [7, 3, 10, 10, Yes], [7, 4, 9, 9, Yes], [7, 5, 8, 8, Yes], [7, 6, 7, 7, Yes], [7, 7, 6, 6, Yes], [7, 8, 5, 5, Yes], [7, 9, 4, 4, Yes], [7, 10, 3, 3, Yes], [7, 11, 2, 2, Yes], [7, 12, 1, 1, Yes], [7, 13, 0, 0, Yes], [7, 14, 19, 19, Yes], [7, 15, 18, 18, Yes], [7, 16, 17, 17, Yes], [7, 17, 16, 16, Yes], [7, 18, 15, 15, Yes], [7, 19, 14, 14, Yes], [8, 0, 12, 12, Yes], [8, 1, 11, 11, Yes], [8, 2, 10, 10, Yes], [8, 3, 9, 9, Yes], [8, 4, 8, 8, Yes], [8, 5, 7, 7, Yes], [8, 6, 6, 6, Yes], [8, 7, 5, 5, Yes], [8, 8, 4, 4, Yes], [8, 9, 3, 3, Yes], [8, 10, 2, 2, Yes], [8, 11, 1, 1, Yes], [8, 12, 0, 0, Yes], [8, 13, 19, 19, Yes], [8, 14, 18, 18, Yes], [8, 15, 17, 17, Yes], [8, 16, 16, 16, Yes], [8, 17, 15, 15, Yes], [8, 18, 14, 14, Yes], [8, 19, 13, 13, Yes], [9, 0, 11, 11, Yes], [9, 1, 10, 10, Yes], [9, 2, 9, 9, Yes], [9, 3, 8, 8, Yes], [9, 4, 7, 7, Yes], [9, 5, 6, 6, Yes], [9, 6, 5, 5, Yes], [9, 7, 4, 4, Yes], [9, 8, 3, 3, Yes], [9, 9, 2, 2, Yes], [9, 10, 1, 1, Yes], [9, 11, 0, 0, Yes], [9, 12, 19, 19, Yes], [9, 13, 18, 18, Yes], [9, 14, 17, 17, Yes], [9, 15, 16, 16, Yes], [9, 16, 15, 15, Yes], [9, 17, 14, 14, Yes], [9, 18, 13, 13, Yes], [9, 19, 12, 12, Yes], [10, 0, 10, 10, Yes], [10, 1, 9, 9, Yes], [10, 2, 8, 8, Yes], [10, 3, 7, 7, Yes], [10, 4, 6, 6, Yes], [10, 5, 5, 5, Yes], [10, 6, 4, 4, Yes], [10, 7, 3, 3, Yes], [10, 8, 2, 2, Yes], [10, 9, 1, 1, Yes], [10, 10, 0, 0, Yes], [10, 11, 19, 19, Yes], [10, 12, 18, 18, Yes], [10, 13, 17, 17, Yes], [10, 14, 16, 16, Yes], [10, 15, 15, 15, Yes], [10, 16, 14, 14, Yes], [10, 17, 13, 13, Yes], [10, 18, 12, 12, Yes], [10, 19, 11, 11, Yes], [11, 0, 9, 9, Yes], [11, 1, 8, 8, Yes], [11, 2, 7, 7, Yes], [11, 3, 6, 6, Yes], [11, 4, 5, 5, Yes], [11, 5, 4, 4, Yes], [11, 6, 3, 3, Yes], [11, 7, 2, 2, Yes], [11, 8, 1, 1, Yes], [11, 9, 0, 0, Yes], [11, 10, 19, 19, Yes], [11, 11, 18, 18, Yes], [11, 12, 17, 17, Yes], [11, 13, 16, 16, Yes], [11, 14, 15, 15, Yes], [11, 15, 14, 14, Yes], [11, 16, 13, 13, Yes], [11, 17, 12, 12, Yes], [11, 18, 11, 11, Yes], [11, 19, 10, 10, Yes], [12, 0, 8, 8, Yes], [12, 1, 7, 7, Yes], [12, 2, 6, 6, Yes], [12, 3, 5, 5, Yes], [12, 4, 4, 4, Yes], [12, 5, 3, 3, Yes], [12, 6, 2, 2, Yes], [12, 7, 1, 1, Yes], [12, 8, 0, 0, Yes], [12, 9, 19, 19, Yes], [12, 10, 18, 18, Yes], [12, 11, 17, 17, Yes], [12, 12, 16, 16, Yes], [12, 13, 15, 15, Yes], [12, 14, 14, 14, Yes], [12, 15, 13, 13, Yes], [12, 16, 12, 12, Yes], [12, 17, 11, 11, Yes], [12, 18, 10, 10, Yes], [12, 19, 9, 9, Yes], [13, 0, 7, 7, Yes], [13, 1, 6, 6, Yes], [13, 2, 5, 5, Yes], [13, 3, 4, 4, Yes], [13, 4, 3, 3, Yes], [13, 5, 2, 2, Yes], [13, 6, 1, 1, Yes], [13, 7, 0, 0, Yes], [13, 8, 19, 19, Yes], [13, 9, 18, 18, Yes], [13, 10, 17, 17, Yes], [13, 11, 16, 16, Yes], [13, 12, 15, 15, Yes], [13, 13, 14, 14, Yes], [13, 14, 13, 13, Yes], [13, 15, 12, 12, Yes], [13, 16, 11, 11, Yes], [13, 17, 10, 10, Yes], [13, 18, 9, 9, Yes], [13, 19, 8, 8, Yes], [14, 0, 6, 6, Yes], [14, 1, 5, 5, Yes], [14, 2, 4, 4, Yes], [14, 3, 3, 3, Yes], [14, 4, 2, 2, Yes], [14, 5, 1, 1, Yes], [14, 6, 0, 0, Yes], [14, 7, 19, 19, Yes], [14, 8, 18, 18, Yes], [14, 9, 17, 17, Yes], [14, 10, 16, 16, Yes], [14, 11, 15, 15, Yes], [14, 12, 14, 14, Yes], [14, 13, 13, 13, Yes], [14, 14, 12, 12, Yes], [14, 15, 11, 11, Yes], [14, 16, 10, 10, Yes], [14, 17, 9, 9, Yes], [14, 18, 8, 8, Yes], [14, 19, 7, 7, Yes], [15, 0, 5, 5, Yes], [15, 1, 4, 4, Yes], [15, 2, 3, 3, Yes], [15, 3, 2, 2, Yes], [15, 4, 1, 1, Yes], [15, 5, 0, 0, Yes], [15, 6, 19, 19, Yes], [15, 7, 18, 18, Yes], [15, 8, 17, 17, Yes], [15, 9, 16, 16, Yes], [15, 10, 15, 15, Yes], [15, 11, 14, 14, Yes], [15, 12, 13, 13, Yes], [15, 13, 12, 12, Yes], [15, 14, 11, 11, Yes], [15, 15, 10, 10, Yes], [15, 16, 9, 9, Yes], [15, 17, 8, 8, Yes], [15, 18, 7, 7, Yes], [15, 19, 6, 6, Yes], [16, 0, 4, 4, Yes], [16, 1, 3, 3, Yes], [16, 2, 2, 2, Yes], [16, 3, 1, 1, Yes], [16, 4, 0, 0, Yes], [16, 5, 19, 19, Yes], [16, 6, 18, 18, Yes], [16, 7, 17, 17, Yes], [16, 8, 16, 16, Yes], [16, 9, 15, 15, Yes], [16, 10, 14, 14, Yes], [16, 11, 13, 13, Yes], [16, 12, 12, 12, Yes], [16, 13, 11, 11, Yes], [16, 14, 10, 10, Yes], [16, 15, 9, 9, Yes], [16, 16, 8, 8, Yes], [16, 17, 7, 7, Yes], [16, 18, 6, 6, Yes], [16, 19, 5, 5, Yes], [17, 0, 3, 3, Yes], [17, 1, 2, 2, Yes], [17, 2, 1, 1, Yes], [17, 3, 0, 0, Yes], [17, 4, 19, 19, Yes], [17, 5, 18, 18, Yes], [17, 6, 17, 17, Yes], [17, 7, 16, 16, Yes], [17, 8, 15, 15, Yes], [17, 9, 14, 14, Yes], [17, 10, 13, 13, Yes], [17, 11, 12, 12, Yes], [17, 12, 11, 11, Yes], [17, 13, 10, 10, Yes], [17, 14, 9, 9, Yes], [17, 15, 8, 8, Yes], [17, 16, 7, 7, Yes], [17, 17, 6, 6, Yes], [17, 18, 5, 5, Yes], [17, 19, 4, 4, Yes], [18, 0, 2, 2, Yes], [18, 1, 1, 1, Yes], [18, 2, 0, 0, Yes], [18, 3, 19, 19, Yes], [18, 4, 18, 18, Yes], [18, 5, 17, 17, Yes], [18, 6, 16, 16, Yes], [18, 7, 15, 15, Yes], [18, 8, 14, 14, Yes], [18, 9, 13, 13, Yes], [18, 10, 12, 12, Yes], [18, 11, 11, 11, Yes], [18, 12, 10, 10, Yes], [18, 13, 9, 9, Yes], [18, 14, 8, 8, Yes], [18, 15, 7, 7, Yes], [18, 16, 6, 6, Yes], [18, 17, 5, 5, Yes], [18, 18, 4, 4, Yes], [18, 19, 3, 3, Yes], [19, 0, 1, 1, Yes], [19, 1, 0, 0, Yes], [19, 2, 19, 19, Yes], [19, 3, 18, 18, Yes], [19, 4, 17, 17, Yes], [19, 5, 16, 16, Yes], [19, 6, 15, 15, Yes], [19, 7, 14, 14, Yes], [19, 8, 13, 13, Yes], [19, 9, 12, 12, Yes], [19, 10, 11, 11, Yes], [19, 11, 10, 10, Yes], [19, 12, 9, 9, Yes], [19, 13, 8, 8, Yes], [19, 14, 7, 7, Yes], [19, 15, 6, 6, Yes], [19, 16, 5, 5, Yes], [19, 17, 4, 4, Yes], [19, 18, 3, 3, Yes], [19, 19, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)
List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (20, 20, [0, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 19, 19, Yes], [0, 2, 18, 18, Yes], [0, 3, 17, 17, Yes], [0, 4, 16, 16, Yes], [0, 5, 15, 15, Yes], [0, 6, 14, 14, Yes], [0, 7, 13, 13, Yes], [0, 8, 12, 12, Yes], [0, 9, 11, 11, Yes], [0, 10, 10, 10, Yes], [0, 11, 9, 9, Yes], [0, 12, 8, 8, Yes], [0, 13, 7, 7, Yes], [0, 14, 6, 6, Yes], [0, 15, 5, 5, Yes], [0, 16, 4, 4, Yes], [0, 17, 3, 3, Yes], [0, 18, 2, 2, Yes], [0, 19, 1, 1, Yes], [1, 0, 19, 19, Yes], [1, 1, 18, 18, Yes], [1, 2, 17, 17, Yes], [1, 3, 16, 16, Yes], [1, 4, 15, 15, Yes], [1, 5, 14, 14, Yes], [1, 6, 13, 13, Yes], [1, 7, 12, 12, Yes], [1, 8, 11, 11, Yes], [1, 9, 10, 10, Yes], [1, 10, 9, 9, Yes], [1, 11, 8, 8, Yes], [1, 12, 7, 7, Yes], [1, 13, 6, 6, Yes], [1, 14, 5, 5, Yes], [1, 15, 4, 4, Yes], [1, 16, 3, 3, Yes], [1, 17, 2, 2, Yes], [1, 18, 1, 1, Yes], [1, 19, 0, 0, Yes], [2, 0, 18, 18, Yes], [2, 1, 17, 17, Yes], [2, 2, 16, 16, Yes], [2, 3, 15, 15, Yes], [2, 4, 14, 14, Yes], [2, 5, 13, 13, Yes], [2, 6, 12, 12, Yes], [2, 7, 11, 11, Yes], [2, 8, 10, 10, Yes], [2, 9, 9, 9, Yes], [2, 10, 8, 8, Yes], [2, 11, 7, 7, Yes], [2, 12, 6, 6, Yes], [2, 13, 5, 5, Yes], [2, 14, 4, 4, Yes], [2, 15, 3, 3, Yes], [2, 16, 2, 2, Yes], [2, 17, 1, 1, Yes], [2, 18, 0, 0, Yes], [2, 19, 19, 19, Yes], [3, 0, 17, 17, Yes], [3, 1, 16, 16, Yes], [3, 2, 15, 15, Yes], [3, 3, 14, 14, Yes], [3, 4, 13, 13, Yes], [3, 5, 12, 12, Yes], [3, 6, 11, 11, Yes], [3, 7, 10, 10, Yes], [3, 8, 9, 9, Yes], [3, 9, 8, 8, Yes], [3, 10, 7, 7, Yes], [3, 11, 6, 6, Yes], [3, 12, 5, 5, Yes], [3, 13, 4, 4, Yes], [3, 14, 3, 3, Yes], [3, 15, 2, 2, Yes], [3, 16, 1, 1, Yes], [3, 17, 0, 0, Yes], [3, 18, 19, 19, Yes], [3, 19, 18, 18, Yes], [4, 0, 16, 16, Yes], [4, 1, 15, 15, Yes], [4, 2, 14, 14, Yes], [4, 3, 13, 13, Yes], [4, 4, 12, 12, Yes], [4, 5, 11, 11, Yes], [4, 6, 10, 10, Yes], [4, 7, 9, 9, Yes], [4, 8, 8, 8, Yes], [4, 9, 7, 7, Yes], [4, 10, 6, 6, Yes], [4, 11, 5, 5, Yes], [4, 12, 4, 4, Yes], [4, 13, 3, 3, Yes], [4, 14, 2, 2, Yes], [4, 15, 1, 1, Yes], [4, 16, 0, 0, Yes], [4, 17, 19, 19, Yes], [4, 18, 18, 18, Yes], [4, 19, 17, 17, Yes], [5, 0, 15, 15, Yes], [5, 1, 14, 14, Yes], [5, 2, 13, 13, Yes], [5, 3, 12, 12, Yes], [5, 4, 11, 11, Yes], [5, 5, 10, 10, Yes], [5, 6, 9, 9, Yes], [5, 7, 8, 8, Yes], [5, 8, 7, 7, Yes], [5, 9, 6, 6, Yes], [5, 10, 5, 5, Yes], [5, 11, 4, 4, Yes], [5, 12, 3, 3, Yes], [5, 13, 2, 2, Yes], [5, 14, 1, 1, Yes], [5, 15, 0, 0, Yes], [5, 16, 19, 19, Yes], [5, 17, 18, 18, Yes], [5, 18, 17, 17, Yes], [5, 19, 16, 16, Yes], [6, 0, 14, 14, Yes], [6, 1, 13, 13, Yes], [6, 2, 12, 12, Yes], [6, 3, 11, 11, Yes], [6, 4, 10, 10, Yes], [6, 5, 9, 9, Yes], [6, 6, 8, 8, Yes], [6, 7, 7, 7, Yes], [6, 8, 6, 6, Yes], [6, 9, 5, 5, Yes], [6, 10, 4, 4, Yes], [6, 11, 3, 3, Yes], [6, 12, 2, 2, Yes], [6, 13, 1, 1, Yes], [6, 14, 0, 0, Yes], [6, 15, 19, 19, Yes], [6, 16, 18, 18, Yes], [6, 17, 17, 17, Yes], [6, 18, 16, 16, Yes], [6, 19, 15, 15, Yes], [7, 0, 13, 13, Yes], [7, 1, 12, 12, Yes], [7, 2, 11, 11, Yes], [7, 3, 10, 10, Yes], [7, 4, 9, 9, Yes], [7, 5, 8, 8, Yes], [7, 6, 7, 7, Yes], [7, 7, 6, 6, Yes], [7, 8, 5, 5, Yes], [7, 9, 4, 4, Yes], [7, 10, 3, 3, Yes], [7, 11, 2, 2, Yes], [7, 12, 1, 1, Yes], [7, 13, 0, 0, Yes], [7, 14, 19, 19, Yes], [7, 15, 18, 18, Yes], [7, 16, 17, 17, Yes], [7, 17, 16, 16, Yes], [7, 18, 15, 15, Yes], [7, 19, 14, 14, Yes], [8, 0, 12, 12, Yes], [8, 1, 11, 11, Yes], [8, 2, 10, 10, Yes], [8, 3, 9, 9, Yes], [8, 4, 8, 8, Yes], [8, 5, 7, 7, Yes], [8, 6, 6, 6, Yes], [8, 7, 5, 5, Yes], [8, 8, 4, 4, Yes], [8, 9, 3, 3, Yes], [8, 10, 2, 2, Yes], [8, 11, 1, 1, Yes], [8, 12, 0, 0, Yes], [8, 13, 19, 19, Yes], [8, 14, 18, 18, Yes], [8, 15, 17, 17, Yes], [8, 16, 16, 16, Yes], [8, 17, 15, 15, Yes], [8, 18, 14, 14, Yes], [8, 19, 13, 13, Yes], [9, 0, 11, 11, Yes], [9, 1, 10, 10, Yes], [9, 2, 9, 9, Yes], [9, 3, 8, 8, Yes], [9, 4, 7, 7, Yes], [9, 5, 6, 6, Yes], [9, 6, 5, 5, Yes], [9, 7, 4, 4, Yes], [9, 8, 3, 3, Yes], [9, 9, 2, 2, Yes], [9, 10, 1, 1, Yes], [9, 11, 0, 0, Yes], [9, 12, 19, 19, Yes], [9, 13, 18, 18, Yes], [9, 14, 17, 17, Yes], [9, 15, 16, 16, Yes], [9, 16, 15, 15, Yes], [9, 17, 14, 14, Yes], [9, 18, 13, 13, Yes], [9, 19, 12, 12, Yes], [10, 0, 10, 10, Yes], [10, 1, 9, 9, Yes], [10, 2, 8, 8, Yes], [10, 3, 7, 7, Yes], [10, 4, 6, 6, Yes], [10, 5, 5, 5, Yes], [10, 6, 4, 4, Yes], [10, 7, 3, 3, Yes], [10, 8, 2, 2, Yes], [10, 9, 1, 1, Yes], [10, 10, 0, 0, Yes], [10, 11, 19, 19, Yes], [10, 12, 18, 18, Yes], [10, 13, 17, 17, Yes], [10, 14, 16, 16, Yes], [10, 15, 15, 15, Yes], [10, 16, 14, 14, Yes], [10, 17, 13, 13, Yes], [10, 18, 12, 12, Yes], [10, 19, 11, 11, Yes], [11, 0, 9, 9, Yes], [11, 1, 8, 8, Yes], [11, 2, 7, 7, Yes], [11, 3, 6, 6, Yes], [11, 4, 5, 5, Yes], [11, 5, 4, 4, Yes], [11, 6, 3, 3, Yes], [11, 7, 2, 2, Yes], [11, 8, 1, 1, Yes], [11, 9, 0, 0, Yes], [11, 10, 19, 19, Yes], [11, 11, 18, 18, Yes], [11, 12, 17, 17, Yes], [11, 13, 16, 16, Yes], [11, 14, 15, 15, Yes], [11, 15, 14, 14, Yes], [11, 16, 13, 13, Yes], [11, 17, 12, 12, Yes], [11, 18, 11, 11, Yes], [11, 19, 10, 10, Yes], [12, 0, 8, 8, Yes], [12, 1, 7, 7, Yes], [12, 2, 6, 6, Yes], [12, 3, 5, 5, Yes], [12, 4, 4, 4, Yes], [12, 5, 3, 3, Yes], [12, 6, 2, 2, Yes], [12, 7, 1, 1, Yes], [12, 8, 0, 0, Yes], [12, 9, 19, 19, Yes], [12, 10, 18, 18, Yes], [12, 11, 17, 17, Yes], [12, 12, 16, 16, Yes], [12, 13, 15, 15, Yes], [12, 14, 14, 14, Yes], [12, 15, 13, 13, Yes], [12, 16, 12, 12, Yes], [12, 17, 11, 11, Yes], [12, 18, 10, 10, Yes], [12, 19, 9, 9, Yes], [13, 0, 7, 7, Yes], [13, 1, 6, 6, Yes], [13, 2, 5, 5, Yes], [13, 3, 4, 4, Yes], [13, 4, 3, 3, Yes], [13, 5, 2, 2, Yes], [13, 6, 1, 1, Yes], [13, 7, 0, 0, Yes], [13, 8, 19, 19, Yes], [13, 9, 18, 18, Yes], [13, 10, 17, 17, Yes], [13, 11, 16, 16, Yes], [13, 12, 15, 15, Yes], [13, 13, 14, 14, Yes], [13, 14, 13, 13, Yes], [13, 15, 12, 12, Yes], [13, 16, 11, 11, Yes], [13, 17, 10, 10, Yes], [13, 18, 9, 9, Yes], [13, 19, 8, 8, Yes], [14, 0, 6, 6, Yes], [14, 1, 5, 5, Yes], [14, 2, 4, 4, Yes], [14, 3, 3, 3, Yes], [14, 4, 2, 2, Yes], [14, 5, 1, 1, Yes], [14, 6, 0, 0, Yes], [14, 7, 19, 19, Yes], [14, 8, 18, 18, Yes], [14, 9, 17, 17, Yes], [14, 10, 16, 16, Yes], [14, 11, 15, 15, Yes], [14, 12, 14, 14, Yes], [14, 13, 13, 13, Yes], [14, 14, 12, 12, Yes], [14, 15, 11, 11, Yes], [14, 16, 10, 10, Yes], [14, 17, 9, 9, Yes], [14, 18, 8, 8, Yes], [14, 19, 7, 7, Yes], [15, 0, 5, 5, Yes], [15, 1, 4, 4, Yes], [15, 2, 3, 3, Yes], [15, 3, 2, 2, Yes], [15, 4, 1, 1, Yes], [15, 5, 0, 0, Yes], [15, 6, 19, 19, Yes], [15, 7, 18, 18, Yes], [15, 8, 17, 17, Yes], [15, 9, 16, 16, Yes], [15, 10, 15, 15, Yes], [15, 11, 14, 14, Yes], [15, 12, 13, 13, Yes], [15, 13, 12, 12, Yes], [15, 14, 11, 11, Yes], [15, 15, 10, 10, Yes], [15, 16, 9, 9, Yes], [15, 17, 8, 8, Yes], [15, 18, 7, 7, Yes], [15, 19, 6, 6, Yes], [16, 0, 4, 4, Yes], [16, 1, 3, 3, Yes], [16, 2, 2, 2, Yes], [16, 3, 1, 1, Yes], [16, 4, 0, 0, Yes], [16, 5, 19, 19, Yes], [16, 6, 18, 18, Yes], [16, 7, 17, 17, Yes], [16, 8, 16, 16, Yes], [16, 9, 15, 15, Yes], [16, 10, 14, 14, Yes], [16, 11, 13, 13, Yes], [16, 12, 12, 12, Yes], [16, 13, 11, 11, Yes], [16, 14, 10, 10, Yes], [16, 15, 9, 9, Yes], [16, 16, 8, 8, Yes], [16, 17, 7, 7, Yes], [16, 18, 6, 6, Yes], [16, 19, 5, 5, Yes], [17, 0, 3, 3, Yes], [17, 1, 2, 2, Yes], [17, 2, 1, 1, Yes], [17, 3, 0, 0, Yes], [17, 4, 19, 19, Yes], [17, 5, 18, 18, Yes], [17, 6, 17, 17, Yes], [17, 7, 16, 16, Yes], [17, 8, 15, 15, Yes], [17, 9, 14, 14, Yes], [17, 10, 13, 13, Yes], [17, 11, 12, 12, Yes], [17, 12, 11, 11, Yes], [17, 13, 10, 10, Yes], [17, 14, 9, 9, Yes], [17, 15, 8, 8, Yes], [17, 16, 7, 7, Yes], [17, 17, 6, 6, Yes], [17, 18, 5, 5, Yes], [17, 19, 4, 4, Yes], [18, 0, 2, 2, Yes], [18, 1, 1, 1, Yes], [18, 2, 0, 0, Yes], [18, 3, 19, 19, Yes], [18, 4, 18, 18, Yes], [18, 5, 17, 17, Yes], [18, 6, 16, 16, Yes], [18, 7, 15, 15, Yes], [18, 8, 14, 14, Yes], [18, 9, 13, 13, Yes], [18, 10, 12, 12, Yes], [18, 11, 11, 11, Yes], [18, 12, 10, 10, Yes], [18, 13, 9, 9, Yes], [18, 14, 8, 8, Yes], [18, 15, 7, 7, Yes], [18, 16, 6, 6, Yes], [18, 17, 5, 5, Yes], [18, 18, 4, 4, Yes], [18, 19, 3, 3, Yes], [19, 0, 1, 1, Yes], [19, 1, 0, 0, Yes], [19, 2, 19, 19, Yes], [19, 3, 18, 18, Yes], [19, 4, 17, 17, Yes], [19, 5, 16, 16, Yes], [19, 6, 15, 15, Yes], [19, 7, 14, 14, Yes], [19, 8, 13, 13, Yes], [19, 9, 12, 12, Yes], [19, 10, 11, 11, Yes], [19, 11, 10, 10, Yes], [19, 12, 9, 9, Yes], [19, 13, 8, 8, Yes], [19, 14, 7, 7, Yes], [19, 15, 6, 6, Yes], [19, 16, 5, 5, Yes], [19, 17, 4, 4, Yes], [19, 18, 3, 3, Yes], [19, 19, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-homomorphism-qualification · candidate
19. Qualify Both Groups and Every Product Pair before Applying Kernel Theorems
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]], target identity=0, and total mapping=[0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).
View answer and explanation
(21, 21, [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 20, 20, Yes], [0, 2, 19, 19, Yes], [0, 3, 18, 18, Yes], [0, 4, 17, 17, Yes], [0, 5, 16, 16, Yes], [0, 6, 15, 15, Yes], [0, 7, 14, 14, Yes], [0, 8, 13, 13, Yes], [0, 9, 12, 12, Yes], [0, 10, 11, 11, Yes], [0, 11, 10, 10, Yes], [0, 12, 9, 9, Yes], [0, 13, 8, 8, Yes], [0, 14, 7, 7, Yes], [0, 15, 6, 6, Yes], [0, 16, 5, 5, Yes], [0, 17, 4, 4, Yes], [0, 18, 3, 3, Yes], [0, 19, 2, 2, Yes], [0, 20, 1, 1, Yes], [1, 0, 20, 20, Yes], [1, 1, 19, 19, Yes], [1, 2, 18, 18, Yes], [1, 3, 17, 17, Yes], [1, 4, 16, 16, Yes], [1, 5, 15, 15, Yes], [1, 6, 14, 14, Yes], [1, 7, 13, 13, Yes], [1, 8, 12, 12, Yes], [1, 9, 11, 11, Yes], [1, 10, 10, 10, Yes], [1, 11, 9, 9, Yes], [1, 12, 8, 8, Yes], [1, 13, 7, 7, Yes], [1, 14, 6, 6, Yes], [1, 15, 5, 5, Yes], [1, 16, 4, 4, Yes], [1, 17, 3, 3, Yes], [1, 18, 2, 2, Yes], [1, 19, 1, 1, Yes], [1, 20, 0, 0, Yes], [2, 0, 19, 19, Yes], [2, 1, 18, 18, Yes], [2, 2, 17, 17, Yes], [2, 3, 16, 16, Yes], [2, 4, 15, 15, Yes], [2, 5, 14, 14, Yes], [2, 6, 13, 13, Yes], [2, 7, 12, 12, Yes], [2, 8, 11, 11, Yes], [2, 9, 10, 10, Yes], [2, 10, 9, 9, Yes], [2, 11, 8, 8, Yes], [2, 12, 7, 7, Yes], [2, 13, 6, 6, Yes], [2, 14, 5, 5, Yes], [2, 15, 4, 4, Yes], [2, 16, 3, 3, Yes], [2, 17, 2, 2, Yes], [2, 18, 1, 1, Yes], [2, 19, 0, 0, Yes], [2, 20, 20, 20, Yes], [3, 0, 18, 18, Yes], [3, 1, 17, 17, Yes], [3, 2, 16, 16, Yes], [3, 3, 15, 15, Yes], [3, 4, 14, 14, Yes], [3, 5, 13, 13, Yes], [3, 6, 12, 12, Yes], [3, 7, 11, 11, Yes], [3, 8, 10, 10, Yes], [3, 9, 9, 9, Yes], [3, 10, 8, 8, Yes], [3, 11, 7, 7, Yes], [3, 12, 6, 6, Yes], [3, 13, 5, 5, Yes], [3, 14, 4, 4, Yes], [3, 15, 3, 3, Yes], [3, 16, 2, 2, Yes], [3, 17, 1, 1, Yes], [3, 18, 0, 0, Yes], [3, 19, 20, 20, Yes], [3, 20, 19, 19, Yes], [4, 0, 17, 17, Yes], [4, 1, 16, 16, Yes], [4, 2, 15, 15, Yes], [4, 3, 14, 14, Yes], [4, 4, 13, 13, Yes], [4, 5, 12, 12, Yes], [4, 6, 11, 11, Yes], [4, 7, 10, 10, Yes], [4, 8, 9, 9, Yes], [4, 9, 8, 8, Yes], [4, 10, 7, 7, Yes], [4, 11, 6, 6, Yes], [4, 12, 5, 5, Yes], [4, 13, 4, 4, Yes], [4, 14, 3, 3, Yes], [4, 15, 2, 2, Yes], [4, 16, 1, 1, Yes], [4, 17, 0, 0, Yes], [4, 18, 20, 20, Yes], [4, 19, 19, 19, Yes], [4, 20, 18, 18, Yes], [5, 0, 16, 16, Yes], [5, 1, 15, 15, Yes], [5, 2, 14, 14, Yes], [5, 3, 13, 13, Yes], [5, 4, 12, 12, Yes], [5, 5, 11, 11, Yes], [5, 6, 10, 10, Yes], [5, 7, 9, 9, Yes], [5, 8, 8, 8, Yes], [5, 9, 7, 7, Yes], [5, 10, 6, 6, Yes], [5, 11, 5, 5, Yes], [5, 12, 4, 4, Yes], [5, 13, 3, 3, Yes], [5, 14, 2, 2, Yes], [5, 15, 1, 1, Yes], [5, 16, 0, 0, Yes], [5, 17, 20, 20, Yes], [5, 18, 19, 19, Yes], [5, 19, 18, 18, Yes], [5, 20, 17, 17, Yes], [6, 0, 15, 15, Yes], [6, 1, 14, 14, Yes], [6, 2, 13, 13, Yes], [6, 3, 12, 12, Yes], [6, 4, 11, 11, Yes], [6, 5, 10, 10, Yes], [6, 6, 9, 9, Yes], [6, 7, 8, 8, Yes], [6, 8, 7, 7, Yes], [6, 9, 6, 6, Yes], [6, 10, 5, 5, Yes], [6, 11, 4, 4, Yes], [6, 12, 3, 3, Yes], [6, 13, 2, 2, Yes], [6, 14, 1, 1, Yes], [6, 15, 0, 0, Yes], [6, 16, 20, 20, Yes], [6, 17, 19, 19, Yes], [6, 18, 18, 18, Yes], [6, 19, 17, 17, Yes], [6, 20, 16, 16, Yes], [7, 0, 14, 14, Yes], [7, 1, 13, 13, Yes], [7, 2, 12, 12, Yes], [7, 3, 11, 11, Yes], [7, 4, 10, 10, Yes], [7, 5, 9, 9, Yes], [7, 6, 8, 8, Yes], [7, 7, 7, 7, Yes], [7, 8, 6, 6, Yes], [7, 9, 5, 5, Yes], [7, 10, 4, 4, Yes], [7, 11, 3, 3, Yes], [7, 12, 2, 2, Yes], [7, 13, 1, 1, Yes], [7, 14, 0, 0, Yes], [7, 15, 20, 20, Yes], [7, 16, 19, 19, Yes], [7, 17, 18, 18, Yes], [7, 18, 17, 17, Yes], [7, 19, 16, 16, Yes], [7, 20, 15, 15, Yes], [8, 0, 13, 13, Yes], [8, 1, 12, 12, Yes], [8, 2, 11, 11, Yes], [8, 3, 10, 10, Yes], [8, 4, 9, 9, Yes], [8, 5, 8, 8, Yes], [8, 6, 7, 7, Yes], [8, 7, 6, 6, Yes], [8, 8, 5, 5, Yes], [8, 9, 4, 4, Yes], [8, 10, 3, 3, Yes], [8, 11, 2, 2, Yes], [8, 12, 1, 1, Yes], [8, 13, 0, 0, Yes], [8, 14, 20, 20, Yes], [8, 15, 19, 19, Yes], [8, 16, 18, 18, Yes], [8, 17, 17, 17, Yes], [8, 18, 16, 16, Yes], [8, 19, 15, 15, Yes], [8, 20, 14, 14, Yes], [9, 0, 12, 12, Yes], [9, 1, 11, 11, Yes], [9, 2, 10, 10, Yes], [9, 3, 9, 9, Yes], [9, 4, 8, 8, Yes], [9, 5, 7, 7, Yes], [9, 6, 6, 6, Yes], [9, 7, 5, 5, Yes], [9, 8, 4, 4, Yes], [9, 9, 3, 3, Yes], [9, 10, 2, 2, Yes], [9, 11, 1, 1, Yes], [9, 12, 0, 0, Yes], [9, 13, 20, 20, Yes], [9, 14, 19, 19, Yes], [9, 15, 18, 18, Yes], [9, 16, 17, 17, Yes], [9, 17, 16, 16, Yes], [9, 18, 15, 15, Yes], [9, 19, 14, 14, Yes], [9, 20, 13, 13, Yes], [10, 0, 11, 11, Yes], [10, 1, 10, 10, Yes], [10, 2, 9, 9, Yes], [10, 3, 8, 8, Yes], [10, 4, 7, 7, Yes], [10, 5, 6, 6, Yes], [10, 6, 5, 5, Yes], [10, 7, 4, 4, Yes], [10, 8, 3, 3, Yes], [10, 9, 2, 2, Yes], [10, 10, 1, 1, Yes], [10, 11, 0, 0, Yes], [10, 12, 20, 20, Yes], [10, 13, 19, 19, Yes], [10, 14, 18, 18, Yes], [10, 15, 17, 17, Yes], [10, 16, 16, 16, Yes], [10, 17, 15, 15, Yes], [10, 18, 14, 14, Yes], [10, 19, 13, 13, Yes], [10, 20, 12, 12, Yes], [11, 0, 10, 10, Yes], [11, 1, 9, 9, Yes], [11, 2, 8, 8, Yes], [11, 3, 7, 7, Yes], [11, 4, 6, 6, Yes], [11, 5, 5, 5, Yes], [11, 6, 4, 4, Yes], [11, 7, 3, 3, Yes], [11, 8, 2, 2, Yes], [11, 9, 1, 1, Yes], [11, 10, 0, 0, Yes], [11, 11, 20, 20, Yes], [11, 12, 19, 19, Yes], [11, 13, 18, 18, Yes], [11, 14, 17, 17, Yes], [11, 15, 16, 16, Yes], [11, 16, 15, 15, Yes], [11, 17, 14, 14, Yes], [11, 18, 13, 13, Yes], [11, 19, 12, 12, Yes], [11, 20, 11, 11, Yes], [12, 0, 9, 9, Yes], [12, 1, 8, 8, Yes], [12, 2, 7, 7, Yes], [12, 3, 6, 6, Yes], [12, 4, 5, 5, Yes], [12, 5, 4, 4, Yes], [12, 6, 3, 3, Yes], [12, 7, 2, 2, Yes], [12, 8, 1, 1, Yes], [12, 9, 0, 0, Yes], [12, 10, 20, 20, Yes], [12, 11, 19, 19, Yes], [12, 12, 18, 18, Yes], [12, 13, 17, 17, Yes], [12, 14, 16, 16, Yes], [12, 15, 15, 15, Yes], [12, 16, 14, 14, Yes], [12, 17, 13, 13, Yes], [12, 18, 12, 12, Yes], [12, 19, 11, 11, Yes], [12, 20, 10, 10, Yes], [13, 0, 8, 8, Yes], [13, 1, 7, 7, Yes], [13, 2, 6, 6, Yes], [13, 3, 5, 5, Yes], [13, 4, 4, 4, Yes], [13, 5, 3, 3, Yes], [13, 6, 2, 2, Yes], [13, 7, 1, 1, Yes], [13, 8, 0, 0, Yes], [13, 9, 20, 20, Yes], [13, 10, 19, 19, Yes], [13, 11, 18, 18, Yes], [13, 12, 17, 17, Yes], [13, 13, 16, 16, Yes], [13, 14, 15, 15, Yes], [13, 15, 14, 14, Yes], [13, 16, 13, 13, Yes], [13, 17, 12, 12, Yes], [13, 18, 11, 11, Yes], [13, 19, 10, 10, Yes], [13, 20, 9, 9, Yes], [14, 0, 7, 7, Yes], [14, 1, 6, 6, Yes], [14, 2, 5, 5, Yes], [14, 3, 4, 4, Yes], [14, 4, 3, 3, Yes], [14, 5, 2, 2, Yes], [14, 6, 1, 1, Yes], [14, 7, 0, 0, Yes], [14, 8, 20, 20, Yes], [14, 9, 19, 19, Yes], [14, 10, 18, 18, Yes], [14, 11, 17, 17, Yes], [14, 12, 16, 16, Yes], [14, 13, 15, 15, Yes], [14, 14, 14, 14, Yes], [14, 15, 13, 13, Yes], [14, 16, 12, 12, Yes], [14, 17, 11, 11, Yes], [14, 18, 10, 10, Yes], [14, 19, 9, 9, Yes], [14, 20, 8, 8, Yes], [15, 0, 6, 6, Yes], [15, 1, 5, 5, Yes], [15, 2, 4, 4, Yes], [15, 3, 3, 3, Yes], [15, 4, 2, 2, Yes], [15, 5, 1, 1, Yes], [15, 6, 0, 0, Yes], [15, 7, 20, 20, Yes], [15, 8, 19, 19, Yes], [15, 9, 18, 18, Yes], [15, 10, 17, 17, Yes], [15, 11, 16, 16, Yes], [15, 12, 15, 15, Yes], [15, 13, 14, 14, Yes], [15, 14, 13, 13, Yes], [15, 15, 12, 12, Yes], [15, 16, 11, 11, Yes], [15, 17, 10, 10, Yes], [15, 18, 9, 9, Yes], [15, 19, 8, 8, Yes], [15, 20, 7, 7, Yes], [16, 0, 5, 5, Yes], [16, 1, 4, 4, Yes], [16, 2, 3, 3, Yes], [16, 3, 2, 2, Yes], [16, 4, 1, 1, Yes], [16, 5, 0, 0, Yes], [16, 6, 20, 20, Yes], [16, 7, 19, 19, Yes], [16, 8, 18, 18, Yes], [16, 9, 17, 17, Yes], [16, 10, 16, 16, Yes], [16, 11, 15, 15, Yes], [16, 12, 14, 14, Yes], [16, 13, 13, 13, Yes], [16, 14, 12, 12, Yes], [16, 15, 11, 11, Yes], [16, 16, 10, 10, Yes], [16, 17, 9, 9, Yes], [16, 18, 8, 8, Yes], [16, 19, 7, 7, Yes], [16, 20, 6, 6, Yes], [17, 0, 4, 4, Yes], [17, 1, 3, 3, Yes], [17, 2, 2, 2, Yes], [17, 3, 1, 1, Yes], [17, 4, 0, 0, Yes], [17, 5, 20, 20, Yes], [17, 6, 19, 19, Yes], [17, 7, 18, 18, Yes], [17, 8, 17, 17, Yes], [17, 9, 16, 16, Yes], [17, 10, 15, 15, Yes], [17, 11, 14, 14, Yes], [17, 12, 13, 13, Yes], [17, 13, 12, 12, Yes], [17, 14, 11, 11, Yes], [17, 15, 10, 10, Yes], [17, 16, 9, 9, Yes], [17, 17, 8, 8, Yes], [17, 18, 7, 7, Yes], [17, 19, 6, 6, Yes], [17, 20, 5, 5, Yes], [18, 0, 3, 3, Yes], [18, 1, 2, 2, Yes], [18, 2, 1, 1, Yes], [18, 3, 0, 0, Yes], [18, 4, 20, 20, Yes], [18, 5, 19, 19, Yes], [18, 6, 18, 18, Yes], [18, 7, 17, 17, Yes], [18, 8, 16, 16, Yes], [18, 9, 15, 15, Yes], [18, 10, 14, 14, Yes], [18, 11, 13, 13, Yes], [18, 12, 12, 12, Yes], [18, 13, 11, 11, Yes], [18, 14, 10, 10, Yes], [18, 15, 9, 9, Yes], [18, 16, 8, 8, Yes], [18, 17, 7, 7, Yes], [18, 18, 6, 6, Yes], [18, 19, 5, 5, Yes], [18, 20, 4, 4, Yes], [19, 0, 2, 2, Yes], [19, 1, 1, 1, Yes], [19, 2, 0, 0, Yes], [19, 3, 20, 20, Yes], [19, 4, 19, 19, Yes], [19, 5, 18, 18, Yes], [19, 6, 17, 17, Yes], [19, 7, 16, 16, Yes], [19, 8, 15, 15, Yes], [19, 9, 14, 14, Yes], [19, 10, 13, 13, Yes], [19, 11, 12, 12, Yes], [19, 12, 11, 11, Yes], [19, 13, 10, 10, Yes], [19, 14, 9, 9, Yes], [19, 15, 8, 8, Yes], [19, 16, 7, 7, Yes], [19, 17, 6, 6, Yes], [19, 18, 5, 5, Yes], [19, 19, 4, 4, Yes], [19, 20, 3, 3, Yes], [20, 0, 1, 1, Yes], [20, 1, 0, 0, Yes], [20, 2, 20, 20, Yes], [20, 3, 19, 19, Yes], [20, 4, 18, 18, Yes], [20, 5, 17, 17, Yes], [20, 6, 16, 16, Yes], [20, 7, 15, 15, Yes], [20, 8, 14, 14, Yes], [20, 9, 13, 13, Yes], [20, 10, 12, 12, Yes], [20, 11, 11, 11, Yes], [20, 12, 10, 10, Yes], [20, 13, 9, 9, Yes], [20, 14, 8, 8, Yes], [20, 15, 7, 7, Yes], [20, 16, 6, 6, Yes], [20, 17, 5, 5, Yes], [20, 18, 4, 4, Yes], [20, 19, 3, 3, Yes], [20, 20, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)
List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (21, 21, [0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 20, 20, Yes], [0, 2, 19, 19, Yes], [0, 3, 18, 18, Yes], [0, 4, 17, 17, Yes], [0, 5, 16, 16, Yes], [0, 6, 15, 15, Yes], [0, 7, 14, 14, Yes], [0, 8, 13, 13, Yes], [0, 9, 12, 12, Yes], [0, 10, 11, 11, Yes], [0, 11, 10, 10, Yes], [0, 12, 9, 9, Yes], [0, 13, 8, 8, Yes], [0, 14, 7, 7, Yes], [0, 15, 6, 6, Yes], [0, 16, 5, 5, Yes], [0, 17, 4, 4, Yes], [0, 18, 3, 3, Yes], [0, 19, 2, 2, Yes], [0, 20, 1, 1, Yes], [1, 0, 20, 20, Yes], [1, 1, 19, 19, Yes], [1, 2, 18, 18, Yes], [1, 3, 17, 17, Yes], [1, 4, 16, 16, Yes], [1, 5, 15, 15, Yes], [1, 6, 14, 14, Yes], [1, 7, 13, 13, Yes], [1, 8, 12, 12, Yes], [1, 9, 11, 11, Yes], [1, 10, 10, 10, Yes], [1, 11, 9, 9, Yes], [1, 12, 8, 8, Yes], [1, 13, 7, 7, Yes], [1, 14, 6, 6, Yes], [1, 15, 5, 5, Yes], [1, 16, 4, 4, Yes], [1, 17, 3, 3, Yes], [1, 18, 2, 2, Yes], [1, 19, 1, 1, Yes], [1, 20, 0, 0, Yes], [2, 0, 19, 19, Yes], [2, 1, 18, 18, Yes], [2, 2, 17, 17, Yes], [2, 3, 16, 16, Yes], [2, 4, 15, 15, Yes], [2, 5, 14, 14, Yes], [2, 6, 13, 13, Yes], [2, 7, 12, 12, Yes], [2, 8, 11, 11, Yes], [2, 9, 10, 10, Yes], [2, 10, 9, 9, Yes], [2, 11, 8, 8, Yes], [2, 12, 7, 7, Yes], [2, 13, 6, 6, Yes], [2, 14, 5, 5, Yes], [2, 15, 4, 4, Yes], [2, 16, 3, 3, Yes], [2, 17, 2, 2, Yes], [2, 18, 1, 1, Yes], [2, 19, 0, 0, Yes], [2, 20, 20, 20, Yes], [3, 0, 18, 18, Yes], [3, 1, 17, 17, Yes], [3, 2, 16, 16, Yes], [3, 3, 15, 15, Yes], [3, 4, 14, 14, Yes], [3, 5, 13, 13, Yes], [3, 6, 12, 12, Yes], [3, 7, 11, 11, Yes], [3, 8, 10, 10, Yes], [3, 9, 9, 9, Yes], [3, 10, 8, 8, Yes], [3, 11, 7, 7, Yes], [3, 12, 6, 6, Yes], [3, 13, 5, 5, Yes], [3, 14, 4, 4, Yes], [3, 15, 3, 3, Yes], [3, 16, 2, 2, Yes], [3, 17, 1, 1, Yes], [3, 18, 0, 0, Yes], [3, 19, 20, 20, Yes], [3, 20, 19, 19, Yes], [4, 0, 17, 17, Yes], [4, 1, 16, 16, Yes], [4, 2, 15, 15, Yes], [4, 3, 14, 14, Yes], [4, 4, 13, 13, Yes], [4, 5, 12, 12, Yes], [4, 6, 11, 11, Yes], [4, 7, 10, 10, Yes], [4, 8, 9, 9, Yes], [4, 9, 8, 8, Yes], [4, 10, 7, 7, Yes], [4, 11, 6, 6, Yes], [4, 12, 5, 5, Yes], [4, 13, 4, 4, Yes], [4, 14, 3, 3, Yes], [4, 15, 2, 2, Yes], [4, 16, 1, 1, Yes], [4, 17, 0, 0, Yes], [4, 18, 20, 20, Yes], [4, 19, 19, 19, Yes], [4, 20, 18, 18, Yes], [5, 0, 16, 16, Yes], [5, 1, 15, 15, Yes], [5, 2, 14, 14, Yes], [5, 3, 13, 13, Yes], [5, 4, 12, 12, Yes], [5, 5, 11, 11, Yes], [5, 6, 10, 10, Yes], [5, 7, 9, 9, Yes], [5, 8, 8, 8, Yes], [5, 9, 7, 7, Yes], [5, 10, 6, 6, Yes], [5, 11, 5, 5, Yes], [5, 12, 4, 4, Yes], [5, 13, 3, 3, Yes], [5, 14, 2, 2, Yes], [5, 15, 1, 1, Yes], [5, 16, 0, 0, Yes], [5, 17, 20, 20, Yes], [5, 18, 19, 19, Yes], [5, 19, 18, 18, Yes], [5, 20, 17, 17, Yes], [6, 0, 15, 15, Yes], [6, 1, 14, 14, Yes], [6, 2, 13, 13, Yes], [6, 3, 12, 12, Yes], [6, 4, 11, 11, Yes], [6, 5, 10, 10, Yes], [6, 6, 9, 9, Yes], [6, 7, 8, 8, Yes], [6, 8, 7, 7, Yes], [6, 9, 6, 6, Yes], [6, 10, 5, 5, Yes], [6, 11, 4, 4, Yes], [6, 12, 3, 3, Yes], [6, 13, 2, 2, Yes], [6, 14, 1, 1, Yes], [6, 15, 0, 0, Yes], [6, 16, 20, 20, Yes], [6, 17, 19, 19, Yes], [6, 18, 18, 18, Yes], [6, 19, 17, 17, Yes], [6, 20, 16, 16, Yes], [7, 0, 14, 14, Yes], [7, 1, 13, 13, Yes], [7, 2, 12, 12, Yes], [7, 3, 11, 11, Yes], [7, 4, 10, 10, Yes], [7, 5, 9, 9, Yes], [7, 6, 8, 8, Yes], [7, 7, 7, 7, Yes], [7, 8, 6, 6, Yes], [7, 9, 5, 5, Yes], [7, 10, 4, 4, Yes], [7, 11, 3, 3, Yes], [7, 12, 2, 2, Yes], [7, 13, 1, 1, Yes], [7, 14, 0, 0, Yes], [7, 15, 20, 20, Yes], [7, 16, 19, 19, Yes], [7, 17, 18, 18, Yes], [7, 18, 17, 17, Yes], [7, 19, 16, 16, Yes], [7, 20, 15, 15, Yes], [8, 0, 13, 13, Yes], [8, 1, 12, 12, Yes], [8, 2, 11, 11, Yes], [8, 3, 10, 10, Yes], [8, 4, 9, 9, Yes], [8, 5, 8, 8, Yes], [8, 6, 7, 7, Yes], [8, 7, 6, 6, Yes], [8, 8, 5, 5, Yes], [8, 9, 4, 4, Yes], [8, 10, 3, 3, Yes], [8, 11, 2, 2, Yes], [8, 12, 1, 1, Yes], [8, 13, 0, 0, Yes], [8, 14, 20, 20, Yes], [8, 15, 19, 19, Yes], [8, 16, 18, 18, Yes], [8, 17, 17, 17, Yes], [8, 18, 16, 16, Yes], [8, 19, 15, 15, Yes], [8, 20, 14, 14, Yes], [9, 0, 12, 12, Yes], [9, 1, 11, 11, Yes], [9, 2, 10, 10, Yes], [9, 3, 9, 9, Yes], [9, 4, 8, 8, Yes], [9, 5, 7, 7, Yes], [9, 6, 6, 6, Yes], [9, 7, 5, 5, Yes], [9, 8, 4, 4, Yes], [9, 9, 3, 3, Yes], [9, 10, 2, 2, Yes], [9, 11, 1, 1, Yes], [9, 12, 0, 0, Yes], [9, 13, 20, 20, Yes], [9, 14, 19, 19, Yes], [9, 15, 18, 18, Yes], [9, 16, 17, 17, Yes], [9, 17, 16, 16, Yes], [9, 18, 15, 15, Yes], [9, 19, 14, 14, Yes], [9, 20, 13, 13, Yes], [10, 0, 11, 11, Yes], [10, 1, 10, 10, Yes], [10, 2, 9, 9, Yes], [10, 3, 8, 8, Yes], [10, 4, 7, 7, Yes], [10, 5, 6, 6, Yes], [10, 6, 5, 5, Yes], [10, 7, 4, 4, Yes], [10, 8, 3, 3, Yes], [10, 9, 2, 2, Yes], [10, 10, 1, 1, Yes], [10, 11, 0, 0, Yes], [10, 12, 20, 20, Yes], [10, 13, 19, 19, Yes], [10, 14, 18, 18, Yes], [10, 15, 17, 17, Yes], [10, 16, 16, 16, Yes], [10, 17, 15, 15, Yes], [10, 18, 14, 14, Yes], [10, 19, 13, 13, Yes], [10, 20, 12, 12, Yes], [11, 0, 10, 10, Yes], [11, 1, 9, 9, Yes], [11, 2, 8, 8, Yes], [11, 3, 7, 7, Yes], [11, 4, 6, 6, Yes], [11, 5, 5, 5, Yes], [11, 6, 4, 4, Yes], [11, 7, 3, 3, Yes], [11, 8, 2, 2, Yes], [11, 9, 1, 1, Yes], [11, 10, 0, 0, Yes], [11, 11, 20, 20, Yes], [11, 12, 19, 19, Yes], [11, 13, 18, 18, Yes], [11, 14, 17, 17, Yes], [11, 15, 16, 16, Yes], [11, 16, 15, 15, Yes], [11, 17, 14, 14, Yes], [11, 18, 13, 13, Yes], [11, 19, 12, 12, Yes], [11, 20, 11, 11, Yes], [12, 0, 9, 9, Yes], [12, 1, 8, 8, Yes], [12, 2, 7, 7, Yes], [12, 3, 6, 6, Yes], [12, 4, 5, 5, Yes], [12, 5, 4, 4, Yes], [12, 6, 3, 3, Yes], [12, 7, 2, 2, Yes], [12, 8, 1, 1, Yes], [12, 9, 0, 0, Yes], [12, 10, 20, 20, Yes], [12, 11, 19, 19, Yes], [12, 12, 18, 18, Yes], [12, 13, 17, 17, Yes], [12, 14, 16, 16, Yes], [12, 15, 15, 15, Yes], [12, 16, 14, 14, Yes], [12, 17, 13, 13, Yes], [12, 18, 12, 12, Yes], [12, 19, 11, 11, Yes], [12, 20, 10, 10, Yes], [13, 0, 8, 8, Yes], [13, 1, 7, 7, Yes], [13, 2, 6, 6, Yes], [13, 3, 5, 5, Yes], [13, 4, 4, 4, Yes], [13, 5, 3, 3, Yes], [13, 6, 2, 2, Yes], [13, 7, 1, 1, Yes], [13, 8, 0, 0, Yes], [13, 9, 20, 20, Yes], [13, 10, 19, 19, Yes], [13, 11, 18, 18, Yes], [13, 12, 17, 17, Yes], [13, 13, 16, 16, Yes], [13, 14, 15, 15, Yes], [13, 15, 14, 14, Yes], [13, 16, 13, 13, Yes], [13, 17, 12, 12, Yes], [13, 18, 11, 11, Yes], [13, 19, 10, 10, Yes], [13, 20, 9, 9, Yes], [14, 0, 7, 7, Yes], [14, 1, 6, 6, Yes], [14, 2, 5, 5, Yes], [14, 3, 4, 4, Yes], [14, 4, 3, 3, Yes], [14, 5, 2, 2, Yes], [14, 6, 1, 1, Yes], [14, 7, 0, 0, Yes], [14, 8, 20, 20, Yes], [14, 9, 19, 19, Yes], [14, 10, 18, 18, Yes], [14, 11, 17, 17, Yes], [14, 12, 16, 16, Yes], [14, 13, 15, 15, Yes], [14, 14, 14, 14, Yes], [14, 15, 13, 13, Yes], [14, 16, 12, 12, Yes], [14, 17, 11, 11, Yes], [14, 18, 10, 10, Yes], [14, 19, 9, 9, Yes], [14, 20, 8, 8, Yes], [15, 0, 6, 6, Yes], [15, 1, 5, 5, Yes], [15, 2, 4, 4, Yes], [15, 3, 3, 3, Yes], [15, 4, 2, 2, Yes], [15, 5, 1, 1, Yes], [15, 6, 0, 0, Yes], [15, 7, 20, 20, Yes], [15, 8, 19, 19, Yes], [15, 9, 18, 18, Yes], [15, 10, 17, 17, Yes], [15, 11, 16, 16, Yes], [15, 12, 15, 15, Yes], [15, 13, 14, 14, Yes], [15, 14, 13, 13, Yes], [15, 15, 12, 12, Yes], [15, 16, 11, 11, Yes], [15, 17, 10, 10, Yes], [15, 18, 9, 9, Yes], [15, 19, 8, 8, Yes], [15, 20, 7, 7, Yes], [16, 0, 5, 5, Yes], [16, 1, 4, 4, Yes], [16, 2, 3, 3, Yes], [16, 3, 2, 2, Yes], [16, 4, 1, 1, Yes], [16, 5, 0, 0, Yes], [16, 6, 20, 20, Yes], [16, 7, 19, 19, Yes], [16, 8, 18, 18, Yes], [16, 9, 17, 17, Yes], [16, 10, 16, 16, Yes], [16, 11, 15, 15, Yes], [16, 12, 14, 14, Yes], [16, 13, 13, 13, Yes], [16, 14, 12, 12, Yes], [16, 15, 11, 11, Yes], [16, 16, 10, 10, Yes], [16, 17, 9, 9, Yes], [16, 18, 8, 8, Yes], [16, 19, 7, 7, Yes], [16, 20, 6, 6, Yes], [17, 0, 4, 4, Yes], [17, 1, 3, 3, Yes], [17, 2, 2, 2, Yes], [17, 3, 1, 1, Yes], [17, 4, 0, 0, Yes], [17, 5, 20, 20, Yes], [17, 6, 19, 19, Yes], [17, 7, 18, 18, Yes], [17, 8, 17, 17, Yes], [17, 9, 16, 16, Yes], [17, 10, 15, 15, Yes], [17, 11, 14, 14, Yes], [17, 12, 13, 13, Yes], [17, 13, 12, 12, Yes], [17, 14, 11, 11, Yes], [17, 15, 10, 10, Yes], [17, 16, 9, 9, Yes], [17, 17, 8, 8, Yes], [17, 18, 7, 7, Yes], [17, 19, 6, 6, Yes], [17, 20, 5, 5, Yes], [18, 0, 3, 3, Yes], [18, 1, 2, 2, Yes], [18, 2, 1, 1, Yes], [18, 3, 0, 0, Yes], [18, 4, 20, 20, Yes], [18, 5, 19, 19, Yes], [18, 6, 18, 18, Yes], [18, 7, 17, 17, Yes], [18, 8, 16, 16, Yes], [18, 9, 15, 15, Yes], [18, 10, 14, 14, Yes], [18, 11, 13, 13, Yes], [18, 12, 12, 12, Yes], [18, 13, 11, 11, Yes], [18, 14, 10, 10, Yes], [18, 15, 9, 9, Yes], [18, 16, 8, 8, Yes], [18, 17, 7, 7, Yes], [18, 18, 6, 6, Yes], [18, 19, 5, 5, Yes], [18, 20, 4, 4, Yes], [19, 0, 2, 2, Yes], [19, 1, 1, 1, Yes], [19, 2, 0, 0, Yes], [19, 3, 20, 20, Yes], [19, 4, 19, 19, Yes], [19, 5, 18, 18, Yes], [19, 6, 17, 17, Yes], [19, 7, 16, 16, Yes], [19, 8, 15, 15, Yes], [19, 9, 14, 14, Yes], [19, 10, 13, 13, Yes], [19, 11, 12, 12, Yes], [19, 12, 11, 11, Yes], [19, 13, 10, 10, Yes], [19, 14, 9, 9, Yes], [19, 15, 8, 8, Yes], [19, 16, 7, 7, Yes], [19, 17, 6, 6, Yes], [19, 18, 5, 5, Yes], [19, 19, 4, 4, Yes], [19, 20, 3, 3, Yes], [20, 0, 1, 1, Yes], [20, 1, 0, 0, Yes], [20, 2, 20, 20, Yes], [20, 3, 19, 19, Yes], [20, 4, 18, 18, Yes], [20, 5, 17, 17, Yes], [20, 6, 16, 16, Yes], [20, 7, 15, 15, Yes], [20, 8, 14, 14, Yes], [20, 9, 13, 13, Yes], [20, 10, 12, 12, Yes], [20, 11, 11, 11, Yes], [20, 12, 10, 10, Yes], [20, 13, 9, 9, Yes], [20, 14, 8, 8, Yes], [20, 15, 7, 7, Yes], [20, 16, 6, 6, Yes], [20, 17, 5, 5, Yes], [20, 18, 4, 4, Yes], [20, 19, 3, 3, Yes], [20, 20, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-homomorphism-qualification · candidate
20. Qualify Both Groups and Every Product Pair before Applying Kernel Theorems
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]], target identity=0, and total mapping=[0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).
View answer and explanation
(22, 22, [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 21, 21, Yes], [0, 2, 20, 20, Yes], [0, 3, 19, 19, Yes], [0, 4, 18, 18, Yes], [0, 5, 17, 17, Yes], [0, 6, 16, 16, Yes], [0, 7, 15, 15, Yes], [0, 8, 14, 14, Yes], [0, 9, 13, 13, Yes], [0, 10, 12, 12, Yes], [0, 11, 11, 11, Yes], [0, 12, 10, 10, Yes], [0, 13, 9, 9, Yes], [0, 14, 8, 8, Yes], [0, 15, 7, 7, Yes], [0, 16, 6, 6, Yes], [0, 17, 5, 5, Yes], [0, 18, 4, 4, Yes], [0, 19, 3, 3, Yes], [0, 20, 2, 2, Yes], [0, 21, 1, 1, Yes], [1, 0, 21, 21, Yes], [1, 1, 20, 20, Yes], [1, 2, 19, 19, Yes], [1, 3, 18, 18, Yes], [1, 4, 17, 17, Yes], [1, 5, 16, 16, Yes], [1, 6, 15, 15, Yes], [1, 7, 14, 14, Yes], [1, 8, 13, 13, Yes], [1, 9, 12, 12, Yes], [1, 10, 11, 11, Yes], [1, 11, 10, 10, Yes], [1, 12, 9, 9, Yes], [1, 13, 8, 8, Yes], [1, 14, 7, 7, Yes], [1, 15, 6, 6, Yes], [1, 16, 5, 5, Yes], [1, 17, 4, 4, Yes], [1, 18, 3, 3, Yes], [1, 19, 2, 2, Yes], [1, 20, 1, 1, Yes], [1, 21, 0, 0, Yes], [2, 0, 20, 20, Yes], [2, 1, 19, 19, Yes], [2, 2, 18, 18, Yes], [2, 3, 17, 17, Yes], [2, 4, 16, 16, Yes], [2, 5, 15, 15, Yes], [2, 6, 14, 14, Yes], [2, 7, 13, 13, Yes], [2, 8, 12, 12, Yes], [2, 9, 11, 11, Yes], [2, 10, 10, 10, Yes], [2, 11, 9, 9, Yes], [2, 12, 8, 8, Yes], [2, 13, 7, 7, Yes], [2, 14, 6, 6, Yes], [2, 15, 5, 5, Yes], [2, 16, 4, 4, Yes], [2, 17, 3, 3, Yes], [2, 18, 2, 2, Yes], [2, 19, 1, 1, Yes], [2, 20, 0, 0, Yes], [2, 21, 21, 21, Yes], [3, 0, 19, 19, Yes], [3, 1, 18, 18, Yes], [3, 2, 17, 17, Yes], [3, 3, 16, 16, Yes], [3, 4, 15, 15, Yes], [3, 5, 14, 14, Yes], [3, 6, 13, 13, Yes], [3, 7, 12, 12, Yes], [3, 8, 11, 11, Yes], [3, 9, 10, 10, Yes], [3, 10, 9, 9, Yes], [3, 11, 8, 8, Yes], [3, 12, 7, 7, Yes], [3, 13, 6, 6, Yes], [3, 14, 5, 5, Yes], [3, 15, 4, 4, Yes], [3, 16, 3, 3, Yes], [3, 17, 2, 2, Yes], [3, 18, 1, 1, Yes], [3, 19, 0, 0, Yes], [3, 20, 21, 21, Yes], [3, 21, 20, 20, Yes], [4, 0, 18, 18, Yes], [4, 1, 17, 17, Yes], [4, 2, 16, 16, Yes], [4, 3, 15, 15, Yes], [4, 4, 14, 14, Yes], [4, 5, 13, 13, Yes], [4, 6, 12, 12, Yes], [4, 7, 11, 11, Yes], [4, 8, 10, 10, Yes], [4, 9, 9, 9, Yes], [4, 10, 8, 8, Yes], [4, 11, 7, 7, Yes], [4, 12, 6, 6, Yes], [4, 13, 5, 5, Yes], [4, 14, 4, 4, Yes], [4, 15, 3, 3, Yes], [4, 16, 2, 2, Yes], [4, 17, 1, 1, Yes], [4, 18, 0, 0, Yes], [4, 19, 21, 21, Yes], [4, 20, 20, 20, Yes], [4, 21, 19, 19, Yes], [5, 0, 17, 17, Yes], [5, 1, 16, 16, Yes], [5, 2, 15, 15, Yes], [5, 3, 14, 14, Yes], [5, 4, 13, 13, Yes], [5, 5, 12, 12, Yes], [5, 6, 11, 11, Yes], [5, 7, 10, 10, Yes], [5, 8, 9, 9, Yes], [5, 9, 8, 8, Yes], [5, 10, 7, 7, Yes], [5, 11, 6, 6, Yes], [5, 12, 5, 5, Yes], [5, 13, 4, 4, Yes], [5, 14, 3, 3, Yes], [5, 15, 2, 2, Yes], [5, 16, 1, 1, Yes], [5, 17, 0, 0, Yes], [5, 18, 21, 21, Yes], [5, 19, 20, 20, Yes], [5, 20, 19, 19, Yes], [5, 21, 18, 18, Yes], [6, 0, 16, 16, Yes], [6, 1, 15, 15, Yes], [6, 2, 14, 14, Yes], [6, 3, 13, 13, Yes], [6, 4, 12, 12, Yes], [6, 5, 11, 11, Yes], [6, 6, 10, 10, Yes], [6, 7, 9, 9, Yes], [6, 8, 8, 8, Yes], [6, 9, 7, 7, Yes], [6, 10, 6, 6, Yes], [6, 11, 5, 5, Yes], [6, 12, 4, 4, Yes], [6, 13, 3, 3, Yes], [6, 14, 2, 2, Yes], [6, 15, 1, 1, Yes], [6, 16, 0, 0, Yes], [6, 17, 21, 21, Yes], [6, 18, 20, 20, Yes], [6, 19, 19, 19, Yes], [6, 20, 18, 18, Yes], [6, 21, 17, 17, Yes], [7, 0, 15, 15, Yes], [7, 1, 14, 14, Yes], [7, 2, 13, 13, Yes], [7, 3, 12, 12, Yes], [7, 4, 11, 11, Yes], [7, 5, 10, 10, Yes], [7, 6, 9, 9, Yes], [7, 7, 8, 8, Yes], [7, 8, 7, 7, Yes], [7, 9, 6, 6, Yes], [7, 10, 5, 5, Yes], [7, 11, 4, 4, Yes], [7, 12, 3, 3, Yes], [7, 13, 2, 2, Yes], [7, 14, 1, 1, Yes], [7, 15, 0, 0, Yes], [7, 16, 21, 21, Yes], [7, 17, 20, 20, Yes], [7, 18, 19, 19, Yes], [7, 19, 18, 18, Yes], [7, 20, 17, 17, Yes], [7, 21, 16, 16, Yes], [8, 0, 14, 14, Yes], [8, 1, 13, 13, Yes], [8, 2, 12, 12, Yes], [8, 3, 11, 11, Yes], [8, 4, 10, 10, Yes], [8, 5, 9, 9, Yes], [8, 6, 8, 8, Yes], [8, 7, 7, 7, Yes], [8, 8, 6, 6, Yes], [8, 9, 5, 5, Yes], [8, 10, 4, 4, Yes], [8, 11, 3, 3, Yes], [8, 12, 2, 2, Yes], [8, 13, 1, 1, Yes], [8, 14, 0, 0, Yes], [8, 15, 21, 21, Yes], [8, 16, 20, 20, Yes], [8, 17, 19, 19, Yes], [8, 18, 18, 18, Yes], [8, 19, 17, 17, Yes], [8, 20, 16, 16, Yes], [8, 21, 15, 15, Yes], [9, 0, 13, 13, Yes], [9, 1, 12, 12, Yes], [9, 2, 11, 11, Yes], [9, 3, 10, 10, Yes], [9, 4, 9, 9, Yes], [9, 5, 8, 8, Yes], [9, 6, 7, 7, Yes], [9, 7, 6, 6, Yes], [9, 8, 5, 5, Yes], [9, 9, 4, 4, Yes], [9, 10, 3, 3, Yes], [9, 11, 2, 2, Yes], [9, 12, 1, 1, Yes], [9, 13, 0, 0, Yes], [9, 14, 21, 21, Yes], [9, 15, 20, 20, Yes], [9, 16, 19, 19, Yes], [9, 17, 18, 18, Yes], [9, 18, 17, 17, Yes], [9, 19, 16, 16, Yes], [9, 20, 15, 15, Yes], [9, 21, 14, 14, Yes], [10, 0, 12, 12, Yes], [10, 1, 11, 11, Yes], [10, 2, 10, 10, Yes], [10, 3, 9, 9, Yes], [10, 4, 8, 8, Yes], [10, 5, 7, 7, Yes], [10, 6, 6, 6, Yes], [10, 7, 5, 5, Yes], [10, 8, 4, 4, Yes], [10, 9, 3, 3, Yes], [10, 10, 2, 2, Yes], [10, 11, 1, 1, Yes], [10, 12, 0, 0, Yes], [10, 13, 21, 21, Yes], [10, 14, 20, 20, Yes], [10, 15, 19, 19, Yes], [10, 16, 18, 18, Yes], [10, 17, 17, 17, Yes], [10, 18, 16, 16, Yes], [10, 19, 15, 15, Yes], [10, 20, 14, 14, Yes], [10, 21, 13, 13, Yes], [11, 0, 11, 11, Yes], [11, 1, 10, 10, Yes], [11, 2, 9, 9, Yes], [11, 3, 8, 8, Yes], [11, 4, 7, 7, Yes], [11, 5, 6, 6, Yes], [11, 6, 5, 5, Yes], [11, 7, 4, 4, Yes], [11, 8, 3, 3, Yes], [11, 9, 2, 2, Yes], [11, 10, 1, 1, Yes], [11, 11, 0, 0, Yes], [11, 12, 21, 21, Yes], [11, 13, 20, 20, Yes], [11, 14, 19, 19, Yes], [11, 15, 18, 18, Yes], [11, 16, 17, 17, Yes], [11, 17, 16, 16, Yes], [11, 18, 15, 15, Yes], [11, 19, 14, 14, Yes], [11, 20, 13, 13, Yes], [11, 21, 12, 12, Yes], [12, 0, 10, 10, Yes], [12, 1, 9, 9, Yes], [12, 2, 8, 8, Yes], [12, 3, 7, 7, Yes], [12, 4, 6, 6, Yes], [12, 5, 5, 5, Yes], [12, 6, 4, 4, Yes], [12, 7, 3, 3, Yes], [12, 8, 2, 2, Yes], [12, 9, 1, 1, Yes], [12, 10, 0, 0, Yes], [12, 11, 21, 21, Yes], [12, 12, 20, 20, Yes], [12, 13, 19, 19, Yes], [12, 14, 18, 18, Yes], [12, 15, 17, 17, Yes], [12, 16, 16, 16, Yes], [12, 17, 15, 15, Yes], [12, 18, 14, 14, Yes], [12, 19, 13, 13, Yes], [12, 20, 12, 12, Yes], [12, 21, 11, 11, Yes], [13, 0, 9, 9, Yes], [13, 1, 8, 8, Yes], [13, 2, 7, 7, Yes], [13, 3, 6, 6, Yes], [13, 4, 5, 5, Yes], [13, 5, 4, 4, Yes], [13, 6, 3, 3, Yes], [13, 7, 2, 2, Yes], [13, 8, 1, 1, Yes], [13, 9, 0, 0, Yes], [13, 10, 21, 21, Yes], [13, 11, 20, 20, Yes], [13, 12, 19, 19, Yes], [13, 13, 18, 18, Yes], [13, 14, 17, 17, Yes], [13, 15, 16, 16, Yes], [13, 16, 15, 15, Yes], [13, 17, 14, 14, Yes], [13, 18, 13, 13, Yes], [13, 19, 12, 12, Yes], [13, 20, 11, 11, Yes], [13, 21, 10, 10, Yes], [14, 0, 8, 8, Yes], [14, 1, 7, 7, Yes], [14, 2, 6, 6, Yes], [14, 3, 5, 5, Yes], [14, 4, 4, 4, Yes], [14, 5, 3, 3, Yes], [14, 6, 2, 2, Yes], [14, 7, 1, 1, Yes], [14, 8, 0, 0, Yes], [14, 9, 21, 21, Yes], [14, 10, 20, 20, Yes], [14, 11, 19, 19, Yes], [14, 12, 18, 18, Yes], [14, 13, 17, 17, Yes], [14, 14, 16, 16, Yes], [14, 15, 15, 15, Yes], [14, 16, 14, 14, Yes], [14, 17, 13, 13, Yes], [14, 18, 12, 12, Yes], [14, 19, 11, 11, Yes], [14, 20, 10, 10, Yes], [14, 21, 9, 9, Yes], [15, 0, 7, 7, Yes], [15, 1, 6, 6, Yes], [15, 2, 5, 5, Yes], [15, 3, 4, 4, Yes], [15, 4, 3, 3, Yes], [15, 5, 2, 2, Yes], [15, 6, 1, 1, Yes], [15, 7, 0, 0, Yes], [15, 8, 21, 21, Yes], [15, 9, 20, 20, Yes], [15, 10, 19, 19, Yes], [15, 11, 18, 18, Yes], [15, 12, 17, 17, Yes], [15, 13, 16, 16, Yes], [15, 14, 15, 15, Yes], [15, 15, 14, 14, Yes], [15, 16, 13, 13, Yes], [15, 17, 12, 12, Yes], [15, 18, 11, 11, Yes], [15, 19, 10, 10, Yes], [15, 20, 9, 9, Yes], [15, 21, 8, 8, Yes], [16, 0, 6, 6, Yes], [16, 1, 5, 5, Yes], [16, 2, 4, 4, Yes], [16, 3, 3, 3, Yes], [16, 4, 2, 2, Yes], [16, 5, 1, 1, Yes], [16, 6, 0, 0, Yes], [16, 7, 21, 21, Yes], [16, 8, 20, 20, Yes], [16, 9, 19, 19, Yes], [16, 10, 18, 18, Yes], [16, 11, 17, 17, Yes], [16, 12, 16, 16, Yes], [16, 13, 15, 15, Yes], [16, 14, 14, 14, Yes], [16, 15, 13, 13, Yes], [16, 16, 12, 12, Yes], [16, 17, 11, 11, Yes], [16, 18, 10, 10, Yes], [16, 19, 9, 9, Yes], [16, 20, 8, 8, Yes], [16, 21, 7, 7, Yes], [17, 0, 5, 5, Yes], [17, 1, 4, 4, Yes], [17, 2, 3, 3, Yes], [17, 3, 2, 2, Yes], [17, 4, 1, 1, Yes], [17, 5, 0, 0, Yes], [17, 6, 21, 21, Yes], [17, 7, 20, 20, Yes], [17, 8, 19, 19, Yes], [17, 9, 18, 18, Yes], [17, 10, 17, 17, Yes], [17, 11, 16, 16, Yes], [17, 12, 15, 15, Yes], [17, 13, 14, 14, Yes], [17, 14, 13, 13, Yes], [17, 15, 12, 12, Yes], [17, 16, 11, 11, Yes], [17, 17, 10, 10, Yes], [17, 18, 9, 9, Yes], [17, 19, 8, 8, Yes], [17, 20, 7, 7, Yes], [17, 21, 6, 6, Yes], [18, 0, 4, 4, Yes], [18, 1, 3, 3, Yes], [18, 2, 2, 2, Yes], [18, 3, 1, 1, Yes], [18, 4, 0, 0, Yes], [18, 5, 21, 21, Yes], [18, 6, 20, 20, Yes], [18, 7, 19, 19, Yes], [18, 8, 18, 18, Yes], [18, 9, 17, 17, Yes], [18, 10, 16, 16, Yes], [18, 11, 15, 15, Yes], [18, 12, 14, 14, Yes], [18, 13, 13, 13, Yes], [18, 14, 12, 12, Yes], [18, 15, 11, 11, Yes], [18, 16, 10, 10, Yes], [18, 17, 9, 9, Yes], [18, 18, 8, 8, Yes], [18, 19, 7, 7, Yes], [18, 20, 6, 6, Yes], [18, 21, 5, 5, Yes], [19, 0, 3, 3, Yes], [19, 1, 2, 2, Yes], [19, 2, 1, 1, Yes], [19, 3, 0, 0, Yes], [19, 4, 21, 21, Yes], [19, 5, 20, 20, Yes], [19, 6, 19, 19, Yes], [19, 7, 18, 18, Yes], [19, 8, 17, 17, Yes], [19, 9, 16, 16, Yes], [19, 10, 15, 15, Yes], [19, 11, 14, 14, Yes], [19, 12, 13, 13, Yes], [19, 13, 12, 12, Yes], [19, 14, 11, 11, Yes], [19, 15, 10, 10, Yes], [19, 16, 9, 9, Yes], [19, 17, 8, 8, Yes], [19, 18, 7, 7, Yes], [19, 19, 6, 6, Yes], [19, 20, 5, 5, Yes], [19, 21, 4, 4, Yes], [20, 0, 2, 2, Yes], [20, 1, 1, 1, Yes], [20, 2, 0, 0, Yes], [20, 3, 21, 21, Yes], [20, 4, 20, 20, Yes], [20, 5, 19, 19, Yes], [20, 6, 18, 18, Yes], [20, 7, 17, 17, Yes], [20, 8, 16, 16, Yes], [20, 9, 15, 15, Yes], [20, 10, 14, 14, Yes], [20, 11, 13, 13, Yes], [20, 12, 12, 12, Yes], [20, 13, 11, 11, Yes], [20, 14, 10, 10, Yes], [20, 15, 9, 9, Yes], [20, 16, 8, 8, Yes], [20, 17, 7, 7, Yes], [20, 18, 6, 6, Yes], [20, 19, 5, 5, Yes], [20, 20, 4, 4, Yes], [20, 21, 3, 3, Yes], [21, 0, 1, 1, Yes], [21, 1, 0, 0, Yes], [21, 2, 21, 21, Yes], [21, 3, 20, 20, Yes], [21, 4, 19, 19, Yes], [21, 5, 18, 18, Yes], [21, 6, 17, 17, Yes], [21, 7, 16, 16, Yes], [21, 8, 15, 15, Yes], [21, 9, 14, 14, Yes], [21, 10, 13, 13, Yes], [21, 11, 12, 12, Yes], [21, 12, 11, 11, Yes], [21, 13, 10, 10, Yes], [21, 14, 9, 9, Yes], [21, 15, 8, 8, Yes], [21, 16, 7, 7, Yes], [21, 17, 6, 6, Yes], [21, 18, 5, 5, Yes], [21, 19, 4, 4, Yes], [21, 20, 3, 3, Yes], [21, 21, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)
List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (22, 22, [0, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 21, 21, Yes], [0, 2, 20, 20, Yes], [0, 3, 19, 19, Yes], [0, 4, 18, 18, Yes], [0, 5, 17, 17, Yes], [0, 6, 16, 16, Yes], [0, 7, 15, 15, Yes], [0, 8, 14, 14, Yes], [0, 9, 13, 13, Yes], [0, 10, 12, 12, Yes], [0, 11, 11, 11, Yes], [0, 12, 10, 10, Yes], [0, 13, 9, 9, Yes], [0, 14, 8, 8, Yes], [0, 15, 7, 7, Yes], [0, 16, 6, 6, Yes], [0, 17, 5, 5, Yes], [0, 18, 4, 4, Yes], [0, 19, 3, 3, Yes], [0, 20, 2, 2, Yes], [0, 21, 1, 1, Yes], [1, 0, 21, 21, Yes], [1, 1, 20, 20, Yes], [1, 2, 19, 19, Yes], [1, 3, 18, 18, Yes], [1, 4, 17, 17, Yes], [1, 5, 16, 16, Yes], [1, 6, 15, 15, Yes], [1, 7, 14, 14, Yes], [1, 8, 13, 13, Yes], [1, 9, 12, 12, Yes], [1, 10, 11, 11, Yes], [1, 11, 10, 10, Yes], [1, 12, 9, 9, Yes], [1, 13, 8, 8, Yes], [1, 14, 7, 7, Yes], [1, 15, 6, 6, Yes], [1, 16, 5, 5, Yes], [1, 17, 4, 4, Yes], [1, 18, 3, 3, Yes], [1, 19, 2, 2, Yes], [1, 20, 1, 1, Yes], [1, 21, 0, 0, Yes], [2, 0, 20, 20, Yes], [2, 1, 19, 19, Yes], [2, 2, 18, 18, Yes], [2, 3, 17, 17, Yes], [2, 4, 16, 16, Yes], [2, 5, 15, 15, Yes], [2, 6, 14, 14, Yes], [2, 7, 13, 13, Yes], [2, 8, 12, 12, Yes], [2, 9, 11, 11, Yes], [2, 10, 10, 10, Yes], [2, 11, 9, 9, Yes], [2, 12, 8, 8, Yes], [2, 13, 7, 7, Yes], [2, 14, 6, 6, Yes], [2, 15, 5, 5, Yes], [2, 16, 4, 4, Yes], [2, 17, 3, 3, Yes], [2, 18, 2, 2, Yes], [2, 19, 1, 1, Yes], [2, 20, 0, 0, Yes], [2, 21, 21, 21, Yes], [3, 0, 19, 19, Yes], [3, 1, 18, 18, Yes], [3, 2, 17, 17, Yes], [3, 3, 16, 16, Yes], [3, 4, 15, 15, Yes], [3, 5, 14, 14, Yes], [3, 6, 13, 13, Yes], [3, 7, 12, 12, Yes], [3, 8, 11, 11, Yes], [3, 9, 10, 10, Yes], [3, 10, 9, 9, Yes], [3, 11, 8, 8, Yes], [3, 12, 7, 7, Yes], [3, 13, 6, 6, Yes], [3, 14, 5, 5, Yes], [3, 15, 4, 4, Yes], [3, 16, 3, 3, Yes], [3, 17, 2, 2, Yes], [3, 18, 1, 1, Yes], [3, 19, 0, 0, Yes], [3, 20, 21, 21, Yes], [3, 21, 20, 20, Yes], [4, 0, 18, 18, Yes], [4, 1, 17, 17, Yes], [4, 2, 16, 16, Yes], [4, 3, 15, 15, Yes], [4, 4, 14, 14, Yes], [4, 5, 13, 13, Yes], [4, 6, 12, 12, Yes], [4, 7, 11, 11, Yes], [4, 8, 10, 10, Yes], [4, 9, 9, 9, Yes], [4, 10, 8, 8, Yes], [4, 11, 7, 7, Yes], [4, 12, 6, 6, Yes], [4, 13, 5, 5, Yes], [4, 14, 4, 4, Yes], [4, 15, 3, 3, Yes], [4, 16, 2, 2, Yes], [4, 17, 1, 1, Yes], [4, 18, 0, 0, Yes], [4, 19, 21, 21, Yes], [4, 20, 20, 20, Yes], [4, 21, 19, 19, Yes], [5, 0, 17, 17, Yes], [5, 1, 16, 16, Yes], [5, 2, 15, 15, Yes], [5, 3, 14, 14, Yes], [5, 4, 13, 13, Yes], [5, 5, 12, 12, Yes], [5, 6, 11, 11, Yes], [5, 7, 10, 10, Yes], [5, 8, 9, 9, Yes], [5, 9, 8, 8, Yes], [5, 10, 7, 7, Yes], [5, 11, 6, 6, Yes], [5, 12, 5, 5, Yes], [5, 13, 4, 4, Yes], [5, 14, 3, 3, Yes], [5, 15, 2, 2, Yes], [5, 16, 1, 1, Yes], [5, 17, 0, 0, Yes], [5, 18, 21, 21, Yes], [5, 19, 20, 20, Yes], [5, 20, 19, 19, Yes], [5, 21, 18, 18, Yes], [6, 0, 16, 16, Yes], [6, 1, 15, 15, Yes], [6, 2, 14, 14, Yes], [6, 3, 13, 13, Yes], [6, 4, 12, 12, Yes], [6, 5, 11, 11, Yes], [6, 6, 10, 10, Yes], [6, 7, 9, 9, Yes], [6, 8, 8, 8, Yes], [6, 9, 7, 7, Yes], [6, 10, 6, 6, Yes], [6, 11, 5, 5, Yes], [6, 12, 4, 4, Yes], [6, 13, 3, 3, Yes], [6, 14, 2, 2, Yes], [6, 15, 1, 1, Yes], [6, 16, 0, 0, Yes], [6, 17, 21, 21, Yes], [6, 18, 20, 20, Yes], [6, 19, 19, 19, Yes], [6, 20, 18, 18, Yes], [6, 21, 17, 17, Yes], [7, 0, 15, 15, Yes], [7, 1, 14, 14, Yes], [7, 2, 13, 13, Yes], [7, 3, 12, 12, Yes], [7, 4, 11, 11, Yes], [7, 5, 10, 10, Yes], [7, 6, 9, 9, Yes], [7, 7, 8, 8, Yes], [7, 8, 7, 7, Yes], [7, 9, 6, 6, Yes], [7, 10, 5, 5, Yes], [7, 11, 4, 4, Yes], [7, 12, 3, 3, Yes], [7, 13, 2, 2, Yes], [7, 14, 1, 1, Yes], [7, 15, 0, 0, Yes], [7, 16, 21, 21, Yes], [7, 17, 20, 20, Yes], [7, 18, 19, 19, Yes], [7, 19, 18, 18, Yes], [7, 20, 17, 17, Yes], [7, 21, 16, 16, Yes], [8, 0, 14, 14, Yes], [8, 1, 13, 13, Yes], [8, 2, 12, 12, Yes], [8, 3, 11, 11, Yes], [8, 4, 10, 10, Yes], [8, 5, 9, 9, Yes], [8, 6, 8, 8, Yes], [8, 7, 7, 7, Yes], [8, 8, 6, 6, Yes], [8, 9, 5, 5, Yes], [8, 10, 4, 4, Yes], [8, 11, 3, 3, Yes], [8, 12, 2, 2, Yes], [8, 13, 1, 1, Yes], [8, 14, 0, 0, Yes], [8, 15, 21, 21, Yes], [8, 16, 20, 20, Yes], [8, 17, 19, 19, Yes], [8, 18, 18, 18, Yes], [8, 19, 17, 17, Yes], [8, 20, 16, 16, Yes], [8, 21, 15, 15, Yes], [9, 0, 13, 13, Yes], [9, 1, 12, 12, Yes], [9, 2, 11, 11, Yes], [9, 3, 10, 10, Yes], [9, 4, 9, 9, Yes], [9, 5, 8, 8, Yes], [9, 6, 7, 7, Yes], [9, 7, 6, 6, Yes], [9, 8, 5, 5, Yes], [9, 9, 4, 4, Yes], [9, 10, 3, 3, Yes], [9, 11, 2, 2, Yes], [9, 12, 1, 1, Yes], [9, 13, 0, 0, Yes], [9, 14, 21, 21, Yes], [9, 15, 20, 20, Yes], [9, 16, 19, 19, Yes], [9, 17, 18, 18, Yes], [9, 18, 17, 17, Yes], [9, 19, 16, 16, Yes], [9, 20, 15, 15, Yes], [9, 21, 14, 14, Yes], [10, 0, 12, 12, Yes], [10, 1, 11, 11, Yes], [10, 2, 10, 10, Yes], [10, 3, 9, 9, Yes], [10, 4, 8, 8, Yes], [10, 5, 7, 7, Yes], [10, 6, 6, 6, Yes], [10, 7, 5, 5, Yes], [10, 8, 4, 4, Yes], [10, 9, 3, 3, Yes], [10, 10, 2, 2, Yes], [10, 11, 1, 1, Yes], [10, 12, 0, 0, Yes], [10, 13, 21, 21, Yes], [10, 14, 20, 20, Yes], [10, 15, 19, 19, Yes], [10, 16, 18, 18, Yes], [10, 17, 17, 17, Yes], [10, 18, 16, 16, Yes], [10, 19, 15, 15, Yes], [10, 20, 14, 14, Yes], [10, 21, 13, 13, Yes], [11, 0, 11, 11, Yes], [11, 1, 10, 10, Yes], [11, 2, 9, 9, Yes], [11, 3, 8, 8, Yes], [11, 4, 7, 7, Yes], [11, 5, 6, 6, Yes], [11, 6, 5, 5, Yes], [11, 7, 4, 4, Yes], [11, 8, 3, 3, Yes], [11, 9, 2, 2, Yes], [11, 10, 1, 1, Yes], [11, 11, 0, 0, Yes], [11, 12, 21, 21, Yes], [11, 13, 20, 20, Yes], [11, 14, 19, 19, Yes], [11, 15, 18, 18, Yes], [11, 16, 17, 17, Yes], [11, 17, 16, 16, Yes], [11, 18, 15, 15, Yes], [11, 19, 14, 14, Yes], [11, 20, 13, 13, Yes], [11, 21, 12, 12, Yes], [12, 0, 10, 10, Yes], [12, 1, 9, 9, Yes], [12, 2, 8, 8, Yes], [12, 3, 7, 7, Yes], [12, 4, 6, 6, Yes], [12, 5, 5, 5, Yes], [12, 6, 4, 4, Yes], [12, 7, 3, 3, Yes], [12, 8, 2, 2, Yes], [12, 9, 1, 1, Yes], [12, 10, 0, 0, Yes], [12, 11, 21, 21, Yes], [12, 12, 20, 20, Yes], [12, 13, 19, 19, Yes], [12, 14, 18, 18, Yes], [12, 15, 17, 17, Yes], [12, 16, 16, 16, Yes], [12, 17, 15, 15, Yes], [12, 18, 14, 14, Yes], [12, 19, 13, 13, Yes], [12, 20, 12, 12, Yes], [12, 21, 11, 11, Yes], [13, 0, 9, 9, Yes], [13, 1, 8, 8, Yes], [13, 2, 7, 7, Yes], [13, 3, 6, 6, Yes], [13, 4, 5, 5, Yes], [13, 5, 4, 4, Yes], [13, 6, 3, 3, Yes], [13, 7, 2, 2, Yes], [13, 8, 1, 1, Yes], [13, 9, 0, 0, Yes], [13, 10, 21, 21, Yes], [13, 11, 20, 20, Yes], [13, 12, 19, 19, Yes], [13, 13, 18, 18, Yes], [13, 14, 17, 17, Yes], [13, 15, 16, 16, Yes], [13, 16, 15, 15, Yes], [13, 17, 14, 14, Yes], [13, 18, 13, 13, Yes], [13, 19, 12, 12, Yes], [13, 20, 11, 11, Yes], [13, 21, 10, 10, Yes], [14, 0, 8, 8, Yes], [14, 1, 7, 7, Yes], [14, 2, 6, 6, Yes], [14, 3, 5, 5, Yes], [14, 4, 4, 4, Yes], [14, 5, 3, 3, Yes], [14, 6, 2, 2, Yes], [14, 7, 1, 1, Yes], [14, 8, 0, 0, Yes], [14, 9, 21, 21, Yes], [14, 10, 20, 20, Yes], [14, 11, 19, 19, Yes], [14, 12, 18, 18, Yes], [14, 13, 17, 17, Yes], [14, 14, 16, 16, Yes], [14, 15, 15, 15, Yes], [14, 16, 14, 14, Yes], [14, 17, 13, 13, Yes], [14, 18, 12, 12, Yes], [14, 19, 11, 11, Yes], [14, 20, 10, 10, Yes], [14, 21, 9, 9, Yes], [15, 0, 7, 7, Yes], [15, 1, 6, 6, Yes], [15, 2, 5, 5, Yes], [15, 3, 4, 4, Yes], [15, 4, 3, 3, Yes], [15, 5, 2, 2, Yes], [15, 6, 1, 1, Yes], [15, 7, 0, 0, Yes], [15, 8, 21, 21, Yes], [15, 9, 20, 20, Yes], [15, 10, 19, 19, Yes], [15, 11, 18, 18, Yes], [15, 12, 17, 17, Yes], [15, 13, 16, 16, Yes], [15, 14, 15, 15, Yes], [15, 15, 14, 14, Yes], [15, 16, 13, 13, Yes], [15, 17, 12, 12, Yes], [15, 18, 11, 11, Yes], [15, 19, 10, 10, Yes], [15, 20, 9, 9, Yes], [15, 21, 8, 8, Yes], [16, 0, 6, 6, Yes], [16, 1, 5, 5, Yes], [16, 2, 4, 4, Yes], [16, 3, 3, 3, Yes], [16, 4, 2, 2, Yes], [16, 5, 1, 1, Yes], [16, 6, 0, 0, Yes], [16, 7, 21, 21, Yes], [16, 8, 20, 20, Yes], [16, 9, 19, 19, Yes], [16, 10, 18, 18, Yes], [16, 11, 17, 17, Yes], [16, 12, 16, 16, Yes], [16, 13, 15, 15, Yes], [16, 14, 14, 14, Yes], [16, 15, 13, 13, Yes], [16, 16, 12, 12, Yes], [16, 17, 11, 11, Yes], [16, 18, 10, 10, Yes], [16, 19, 9, 9, Yes], [16, 20, 8, 8, Yes], [16, 21, 7, 7, Yes], [17, 0, 5, 5, Yes], [17, 1, 4, 4, Yes], [17, 2, 3, 3, Yes], [17, 3, 2, 2, Yes], [17, 4, 1, 1, Yes], [17, 5, 0, 0, Yes], [17, 6, 21, 21, Yes], [17, 7, 20, 20, Yes], [17, 8, 19, 19, Yes], [17, 9, 18, 18, Yes], [17, 10, 17, 17, Yes], [17, 11, 16, 16, Yes], [17, 12, 15, 15, Yes], [17, 13, 14, 14, Yes], [17, 14, 13, 13, Yes], [17, 15, 12, 12, Yes], [17, 16, 11, 11, Yes], [17, 17, 10, 10, Yes], [17, 18, 9, 9, Yes], [17, 19, 8, 8, Yes], [17, 20, 7, 7, Yes], [17, 21, 6, 6, Yes], [18, 0, 4, 4, Yes], [18, 1, 3, 3, Yes], [18, 2, 2, 2, Yes], [18, 3, 1, 1, Yes], [18, 4, 0, 0, Yes], [18, 5, 21, 21, Yes], [18, 6, 20, 20, Yes], [18, 7, 19, 19, Yes], [18, 8, 18, 18, Yes], [18, 9, 17, 17, Yes], [18, 10, 16, 16, Yes], [18, 11, 15, 15, Yes], [18, 12, 14, 14, Yes], [18, 13, 13, 13, Yes], [18, 14, 12, 12, Yes], [18, 15, 11, 11, Yes], [18, 16, 10, 10, Yes], [18, 17, 9, 9, Yes], [18, 18, 8, 8, Yes], [18, 19, 7, 7, Yes], [18, 20, 6, 6, Yes], [18, 21, 5, 5, Yes], [19, 0, 3, 3, Yes], [19, 1, 2, 2, Yes], [19, 2, 1, 1, Yes], [19, 3, 0, 0, Yes], [19, 4, 21, 21, Yes], [19, 5, 20, 20, Yes], [19, 6, 19, 19, Yes], [19, 7, 18, 18, Yes], [19, 8, 17, 17, Yes], [19, 9, 16, 16, Yes], [19, 10, 15, 15, Yes], [19, 11, 14, 14, Yes], [19, 12, 13, 13, Yes], [19, 13, 12, 12, Yes], [19, 14, 11, 11, Yes], [19, 15, 10, 10, Yes], [19, 16, 9, 9, Yes], [19, 17, 8, 8, Yes], [19, 18, 7, 7, Yes], [19, 19, 6, 6, Yes], [19, 20, 5, 5, Yes], [19, 21, 4, 4, Yes], [20, 0, 2, 2, Yes], [20, 1, 1, 1, Yes], [20, 2, 0, 0, Yes], [20, 3, 21, 21, Yes], [20, 4, 20, 20, Yes], [20, 5, 19, 19, Yes], [20, 6, 18, 18, Yes], [20, 7, 17, 17, Yes], [20, 8, 16, 16, Yes], [20, 9, 15, 15, Yes], [20, 10, 14, 14, Yes], [20, 11, 13, 13, Yes], [20, 12, 12, 12, Yes], [20, 13, 11, 11, Yes], [20, 14, 10, 10, Yes], [20, 15, 9, 9, Yes], [20, 16, 8, 8, Yes], [20, 17, 7, 7, Yes], [20, 18, 6, 6, Yes], [20, 19, 5, 5, Yes], [20, 20, 4, 4, Yes], [20, 21, 3, 3, Yes], [21, 0, 1, 1, Yes], [21, 1, 0, 0, Yes], [21, 2, 21, 21, Yes], [21, 3, 20, 20, Yes], [21, 4, 19, 19, Yes], [21, 5, 18, 18, Yes], [21, 6, 17, 17, Yes], [21, 7, 16, 16, Yes], [21, 8, 15, 15, Yes], [21, 9, 14, 14, Yes], [21, 10, 13, 13, Yes], [21, 11, 12, 12, Yes], [21, 12, 11, 11, Yes], [21, 13, 10, 10, Yes], [21, 14, 9, 9, Yes], [21, 15, 8, 8, Yes], [21, 16, 7, 7, Yes], [21, 17, 6, 6, Yes], [21, 18, 5, 5, Yes], [21, 19, 4, 4, Yes], [21, 20, 3, 3, Yes], [21, 21, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-homomorphism-qualification · candidate
21. Qualify Both Groups and Every Product Pair before Applying Kernel Theorems
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]], target identity=0, and total mapping=[0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).
View answer and explanation
(23, 23, [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 22, 22, Yes], [0, 2, 21, 21, Yes], [0, 3, 20, 20, Yes], [0, 4, 19, 19, Yes], [0, 5, 18, 18, Yes], [0, 6, 17, 17, Yes], [0, 7, 16, 16, Yes], [0, 8, 15, 15, Yes], [0, 9, 14, 14, Yes], [0, 10, 13, 13, Yes], [0, 11, 12, 12, Yes], [0, 12, 11, 11, Yes], [0, 13, 10, 10, Yes], [0, 14, 9, 9, Yes], [0, 15, 8, 8, Yes], [0, 16, 7, 7, Yes], [0, 17, 6, 6, Yes], [0, 18, 5, 5, Yes], [0, 19, 4, 4, Yes], [0, 20, 3, 3, Yes], [0, 21, 2, 2, Yes], [0, 22, 1, 1, Yes], [1, 0, 22, 22, Yes], [1, 1, 21, 21, Yes], [1, 2, 20, 20, Yes], [1, 3, 19, 19, Yes], [1, 4, 18, 18, Yes], [1, 5, 17, 17, Yes], [1, 6, 16, 16, Yes], [1, 7, 15, 15, Yes], [1, 8, 14, 14, Yes], [1, 9, 13, 13, Yes], [1, 10, 12, 12, Yes], [1, 11, 11, 11, Yes], [1, 12, 10, 10, Yes], [1, 13, 9, 9, Yes], [1, 14, 8, 8, Yes], [1, 15, 7, 7, Yes], [1, 16, 6, 6, Yes], [1, 17, 5, 5, Yes], [1, 18, 4, 4, Yes], [1, 19, 3, 3, Yes], [1, 20, 2, 2, Yes], [1, 21, 1, 1, Yes], [1, 22, 0, 0, Yes], [2, 0, 21, 21, Yes], [2, 1, 20, 20, Yes], [2, 2, 19, 19, Yes], [2, 3, 18, 18, Yes], [2, 4, 17, 17, Yes], [2, 5, 16, 16, Yes], [2, 6, 15, 15, Yes], [2, 7, 14, 14, Yes], [2, 8, 13, 13, Yes], [2, 9, 12, 12, Yes], [2, 10, 11, 11, Yes], [2, 11, 10, 10, Yes], [2, 12, 9, 9, Yes], [2, 13, 8, 8, Yes], [2, 14, 7, 7, Yes], [2, 15, 6, 6, Yes], [2, 16, 5, 5, Yes], [2, 17, 4, 4, Yes], [2, 18, 3, 3, Yes], [2, 19, 2, 2, Yes], [2, 20, 1, 1, Yes], [2, 21, 0, 0, Yes], [2, 22, 22, 22, Yes], [3, 0, 20, 20, Yes], [3, 1, 19, 19, Yes], [3, 2, 18, 18, Yes], [3, 3, 17, 17, Yes], [3, 4, 16, 16, Yes], [3, 5, 15, 15, Yes], [3, 6, 14, 14, Yes], [3, 7, 13, 13, Yes], [3, 8, 12, 12, Yes], [3, 9, 11, 11, Yes], [3, 10, 10, 10, Yes], [3, 11, 9, 9, Yes], [3, 12, 8, 8, Yes], [3, 13, 7, 7, Yes], [3, 14, 6, 6, Yes], [3, 15, 5, 5, Yes], [3, 16, 4, 4, Yes], [3, 17, 3, 3, Yes], [3, 18, 2, 2, Yes], [3, 19, 1, 1, Yes], [3, 20, 0, 0, Yes], [3, 21, 22, 22, Yes], [3, 22, 21, 21, Yes], [4, 0, 19, 19, Yes], [4, 1, 18, 18, Yes], [4, 2, 17, 17, Yes], [4, 3, 16, 16, Yes], [4, 4, 15, 15, Yes], [4, 5, 14, 14, Yes], [4, 6, 13, 13, Yes], [4, 7, 12, 12, Yes], [4, 8, 11, 11, Yes], [4, 9, 10, 10, Yes], [4, 10, 9, 9, Yes], [4, 11, 8, 8, Yes], [4, 12, 7, 7, Yes], [4, 13, 6, 6, Yes], [4, 14, 5, 5, Yes], [4, 15, 4, 4, Yes], [4, 16, 3, 3, Yes], [4, 17, 2, 2, Yes], [4, 18, 1, 1, Yes], [4, 19, 0, 0, Yes], [4, 20, 22, 22, Yes], [4, 21, 21, 21, Yes], [4, 22, 20, 20, Yes], [5, 0, 18, 18, Yes], [5, 1, 17, 17, Yes], [5, 2, 16, 16, Yes], [5, 3, 15, 15, Yes], [5, 4, 14, 14, Yes], [5, 5, 13, 13, Yes], [5, 6, 12, 12, Yes], [5, 7, 11, 11, Yes], [5, 8, 10, 10, Yes], [5, 9, 9, 9, Yes], [5, 10, 8, 8, Yes], [5, 11, 7, 7, Yes], [5, 12, 6, 6, Yes], [5, 13, 5, 5, Yes], [5, 14, 4, 4, Yes], [5, 15, 3, 3, Yes], [5, 16, 2, 2, Yes], [5, 17, 1, 1, Yes], [5, 18, 0, 0, Yes], [5, 19, 22, 22, Yes], [5, 20, 21, 21, Yes], [5, 21, 20, 20, Yes], [5, 22, 19, 19, Yes], [6, 0, 17, 17, Yes], [6, 1, 16, 16, Yes], [6, 2, 15, 15, Yes], [6, 3, 14, 14, Yes], [6, 4, 13, 13, Yes], [6, 5, 12, 12, Yes], [6, 6, 11, 11, Yes], [6, 7, 10, 10, Yes], [6, 8, 9, 9, Yes], [6, 9, 8, 8, Yes], [6, 10, 7, 7, Yes], [6, 11, 6, 6, Yes], [6, 12, 5, 5, Yes], [6, 13, 4, 4, Yes], [6, 14, 3, 3, Yes], [6, 15, 2, 2, Yes], [6, 16, 1, 1, Yes], [6, 17, 0, 0, Yes], [6, 18, 22, 22, Yes], [6, 19, 21, 21, Yes], [6, 20, 20, 20, Yes], [6, 21, 19, 19, Yes], [6, 22, 18, 18, Yes], [7, 0, 16, 16, Yes], [7, 1, 15, 15, Yes], [7, 2, 14, 14, Yes], [7, 3, 13, 13, Yes], [7, 4, 12, 12, Yes], [7, 5, 11, 11, Yes], [7, 6, 10, 10, Yes], [7, 7, 9, 9, Yes], [7, 8, 8, 8, Yes], [7, 9, 7, 7, Yes], [7, 10, 6, 6, Yes], [7, 11, 5, 5, Yes], [7, 12, 4, 4, Yes], [7, 13, 3, 3, Yes], [7, 14, 2, 2, Yes], [7, 15, 1, 1, Yes], [7, 16, 0, 0, Yes], [7, 17, 22, 22, Yes], [7, 18, 21, 21, Yes], [7, 19, 20, 20, Yes], [7, 20, 19, 19, Yes], [7, 21, 18, 18, Yes], [7, 22, 17, 17, Yes], [8, 0, 15, 15, Yes], [8, 1, 14, 14, Yes], [8, 2, 13, 13, Yes], [8, 3, 12, 12, Yes], [8, 4, 11, 11, Yes], [8, 5, 10, 10, Yes], [8, 6, 9, 9, Yes], [8, 7, 8, 8, Yes], [8, 8, 7, 7, Yes], [8, 9, 6, 6, Yes], [8, 10, 5, 5, Yes], [8, 11, 4, 4, Yes], [8, 12, 3, 3, Yes], [8, 13, 2, 2, Yes], [8, 14, 1, 1, Yes], [8, 15, 0, 0, Yes], [8, 16, 22, 22, Yes], [8, 17, 21, 21, Yes], [8, 18, 20, 20, Yes], [8, 19, 19, 19, Yes], [8, 20, 18, 18, Yes], [8, 21, 17, 17, Yes], [8, 22, 16, 16, Yes], [9, 0, 14, 14, Yes], [9, 1, 13, 13, Yes], [9, 2, 12, 12, Yes], [9, 3, 11, 11, Yes], [9, 4, 10, 10, Yes], [9, 5, 9, 9, Yes], [9, 6, 8, 8, Yes], [9, 7, 7, 7, Yes], [9, 8, 6, 6, Yes], [9, 9, 5, 5, Yes], [9, 10, 4, 4, Yes], [9, 11, 3, 3, Yes], [9, 12, 2, 2, Yes], [9, 13, 1, 1, Yes], [9, 14, 0, 0, Yes], [9, 15, 22, 22, Yes], [9, 16, 21, 21, Yes], [9, 17, 20, 20, Yes], [9, 18, 19, 19, Yes], [9, 19, 18, 18, Yes], [9, 20, 17, 17, Yes], [9, 21, 16, 16, Yes], [9, 22, 15, 15, Yes], [10, 0, 13, 13, Yes], [10, 1, 12, 12, Yes], [10, 2, 11, 11, Yes], [10, 3, 10, 10, Yes], [10, 4, 9, 9, Yes], [10, 5, 8, 8, Yes], [10, 6, 7, 7, Yes], [10, 7, 6, 6, Yes], [10, 8, 5, 5, Yes], [10, 9, 4, 4, Yes], [10, 10, 3, 3, Yes], [10, 11, 2, 2, Yes], [10, 12, 1, 1, Yes], [10, 13, 0, 0, Yes], [10, 14, 22, 22, Yes], [10, 15, 21, 21, Yes], [10, 16, 20, 20, Yes], [10, 17, 19, 19, Yes], [10, 18, 18, 18, Yes], [10, 19, 17, 17, Yes], [10, 20, 16, 16, Yes], [10, 21, 15, 15, Yes], [10, 22, 14, 14, Yes], [11, 0, 12, 12, Yes], [11, 1, 11, 11, Yes], [11, 2, 10, 10, Yes], [11, 3, 9, 9, Yes], [11, 4, 8, 8, Yes], [11, 5, 7, 7, Yes], [11, 6, 6, 6, Yes], [11, 7, 5, 5, Yes], [11, 8, 4, 4, Yes], [11, 9, 3, 3, Yes], [11, 10, 2, 2, Yes], [11, 11, 1, 1, Yes], [11, 12, 0, 0, Yes], [11, 13, 22, 22, Yes], [11, 14, 21, 21, Yes], [11, 15, 20, 20, Yes], [11, 16, 19, 19, Yes], [11, 17, 18, 18, Yes], [11, 18, 17, 17, Yes], [11, 19, 16, 16, Yes], [11, 20, 15, 15, Yes], [11, 21, 14, 14, Yes], [11, 22, 13, 13, Yes], [12, 0, 11, 11, Yes], [12, 1, 10, 10, Yes], [12, 2, 9, 9, Yes], [12, 3, 8, 8, Yes], [12, 4, 7, 7, Yes], [12, 5, 6, 6, Yes], [12, 6, 5, 5, Yes], [12, 7, 4, 4, Yes], [12, 8, 3, 3, Yes], [12, 9, 2, 2, Yes], [12, 10, 1, 1, Yes], [12, 11, 0, 0, Yes], [12, 12, 22, 22, Yes], [12, 13, 21, 21, Yes], [12, 14, 20, 20, Yes], [12, 15, 19, 19, Yes], [12, 16, 18, 18, Yes], [12, 17, 17, 17, Yes], [12, 18, 16, 16, Yes], [12, 19, 15, 15, Yes], [12, 20, 14, 14, Yes], [12, 21, 13, 13, Yes], [12, 22, 12, 12, Yes], [13, 0, 10, 10, Yes], [13, 1, 9, 9, Yes], [13, 2, 8, 8, Yes], [13, 3, 7, 7, Yes], [13, 4, 6, 6, Yes], [13, 5, 5, 5, Yes], [13, 6, 4, 4, Yes], [13, 7, 3, 3, Yes], [13, 8, 2, 2, Yes], [13, 9, 1, 1, Yes], [13, 10, 0, 0, Yes], [13, 11, 22, 22, Yes], [13, 12, 21, 21, Yes], [13, 13, 20, 20, Yes], [13, 14, 19, 19, Yes], [13, 15, 18, 18, Yes], [13, 16, 17, 17, Yes], [13, 17, 16, 16, Yes], [13, 18, 15, 15, Yes], [13, 19, 14, 14, Yes], [13, 20, 13, 13, Yes], [13, 21, 12, 12, Yes], [13, 22, 11, 11, Yes], [14, 0, 9, 9, Yes], [14, 1, 8, 8, Yes], [14, 2, 7, 7, Yes], [14, 3, 6, 6, Yes], [14, 4, 5, 5, Yes], [14, 5, 4, 4, Yes], [14, 6, 3, 3, Yes], [14, 7, 2, 2, Yes], [14, 8, 1, 1, Yes], [14, 9, 0, 0, Yes], [14, 10, 22, 22, Yes], [14, 11, 21, 21, Yes], [14, 12, 20, 20, Yes], [14, 13, 19, 19, Yes], [14, 14, 18, 18, Yes], [14, 15, 17, 17, Yes], [14, 16, 16, 16, Yes], [14, 17, 15, 15, Yes], [14, 18, 14, 14, Yes], [14, 19, 13, 13, Yes], [14, 20, 12, 12, Yes], [14, 21, 11, 11, Yes], [14, 22, 10, 10, Yes], [15, 0, 8, 8, Yes], [15, 1, 7, 7, Yes], [15, 2, 6, 6, Yes], [15, 3, 5, 5, Yes], [15, 4, 4, 4, Yes], [15, 5, 3, 3, Yes], [15, 6, 2, 2, Yes], [15, 7, 1, 1, Yes], [15, 8, 0, 0, Yes], [15, 9, 22, 22, Yes], [15, 10, 21, 21, Yes], [15, 11, 20, 20, Yes], [15, 12, 19, 19, Yes], [15, 13, 18, 18, Yes], [15, 14, 17, 17, Yes], [15, 15, 16, 16, Yes], [15, 16, 15, 15, Yes], [15, 17, 14, 14, Yes], [15, 18, 13, 13, Yes], [15, 19, 12, 12, Yes], [15, 20, 11, 11, Yes], [15, 21, 10, 10, Yes], [15, 22, 9, 9, Yes], [16, 0, 7, 7, Yes], [16, 1, 6, 6, Yes], [16, 2, 5, 5, Yes], [16, 3, 4, 4, Yes], [16, 4, 3, 3, Yes], [16, 5, 2, 2, Yes], [16, 6, 1, 1, Yes], [16, 7, 0, 0, Yes], [16, 8, 22, 22, Yes], [16, 9, 21, 21, Yes], [16, 10, 20, 20, Yes], [16, 11, 19, 19, Yes], [16, 12, 18, 18, Yes], [16, 13, 17, 17, Yes], [16, 14, 16, 16, Yes], [16, 15, 15, 15, Yes], [16, 16, 14, 14, Yes], [16, 17, 13, 13, Yes], [16, 18, 12, 12, Yes], [16, 19, 11, 11, Yes], [16, 20, 10, 10, Yes], [16, 21, 9, 9, Yes], [16, 22, 8, 8, Yes], [17, 0, 6, 6, Yes], [17, 1, 5, 5, Yes], [17, 2, 4, 4, Yes], [17, 3, 3, 3, Yes], [17, 4, 2, 2, Yes], [17, 5, 1, 1, Yes], [17, 6, 0, 0, Yes], [17, 7, 22, 22, Yes], [17, 8, 21, 21, Yes], [17, 9, 20, 20, Yes], [17, 10, 19, 19, Yes], [17, 11, 18, 18, Yes], [17, 12, 17, 17, Yes], [17, 13, 16, 16, Yes], [17, 14, 15, 15, Yes], [17, 15, 14, 14, Yes], [17, 16, 13, 13, Yes], [17, 17, 12, 12, Yes], [17, 18, 11, 11, Yes], [17, 19, 10, 10, Yes], [17, 20, 9, 9, Yes], [17, 21, 8, 8, Yes], [17, 22, 7, 7, Yes], [18, 0, 5, 5, Yes], [18, 1, 4, 4, Yes], [18, 2, 3, 3, Yes], [18, 3, 2, 2, Yes], [18, 4, 1, 1, Yes], [18, 5, 0, 0, Yes], [18, 6, 22, 22, Yes], [18, 7, 21, 21, Yes], [18, 8, 20, 20, Yes], [18, 9, 19, 19, Yes], [18, 10, 18, 18, Yes], [18, 11, 17, 17, Yes], [18, 12, 16, 16, Yes], [18, 13, 15, 15, Yes], [18, 14, 14, 14, Yes], [18, 15, 13, 13, Yes], [18, 16, 12, 12, Yes], [18, 17, 11, 11, Yes], [18, 18, 10, 10, Yes], [18, 19, 9, 9, Yes], [18, 20, 8, 8, Yes], [18, 21, 7, 7, Yes], [18, 22, 6, 6, Yes], [19, 0, 4, 4, Yes], [19, 1, 3, 3, Yes], [19, 2, 2, 2, Yes], [19, 3, 1, 1, Yes], [19, 4, 0, 0, Yes], [19, 5, 22, 22, Yes], [19, 6, 21, 21, Yes], [19, 7, 20, 20, Yes], [19, 8, 19, 19, Yes], [19, 9, 18, 18, Yes], [19, 10, 17, 17, Yes], [19, 11, 16, 16, Yes], [19, 12, 15, 15, Yes], [19, 13, 14, 14, Yes], [19, 14, 13, 13, Yes], [19, 15, 12, 12, Yes], [19, 16, 11, 11, Yes], [19, 17, 10, 10, Yes], [19, 18, 9, 9, Yes], [19, 19, 8, 8, Yes], [19, 20, 7, 7, Yes], [19, 21, 6, 6, Yes], [19, 22, 5, 5, Yes], [20, 0, 3, 3, Yes], [20, 1, 2, 2, Yes], [20, 2, 1, 1, Yes], [20, 3, 0, 0, Yes], [20, 4, 22, 22, Yes], [20, 5, 21, 21, Yes], [20, 6, 20, 20, Yes], [20, 7, 19, 19, Yes], [20, 8, 18, 18, Yes], [20, 9, 17, 17, Yes], [20, 10, 16, 16, Yes], [20, 11, 15, 15, Yes], [20, 12, 14, 14, Yes], [20, 13, 13, 13, Yes], [20, 14, 12, 12, Yes], [20, 15, 11, 11, Yes], [20, 16, 10, 10, Yes], [20, 17, 9, 9, Yes], [20, 18, 8, 8, Yes], [20, 19, 7, 7, Yes], [20, 20, 6, 6, Yes], [20, 21, 5, 5, Yes], [20, 22, 4, 4, Yes], [21, 0, 2, 2, Yes], [21, 1, 1, 1, Yes], [21, 2, 0, 0, Yes], [21, 3, 22, 22, Yes], [21, 4, 21, 21, Yes], [21, 5, 20, 20, Yes], [21, 6, 19, 19, Yes], [21, 7, 18, 18, Yes], [21, 8, 17, 17, Yes], [21, 9, 16, 16, Yes], [21, 10, 15, 15, Yes], [21, 11, 14, 14, Yes], [21, 12, 13, 13, Yes], [21, 13, 12, 12, Yes], [21, 14, 11, 11, Yes], [21, 15, 10, 10, Yes], [21, 16, 9, 9, Yes], [21, 17, 8, 8, Yes], [21, 18, 7, 7, Yes], [21, 19, 6, 6, Yes], [21, 20, 5, 5, Yes], [21, 21, 4, 4, Yes], [21, 22, 3, 3, Yes], [22, 0, 1, 1, Yes], [22, 1, 0, 0, Yes], [22, 2, 22, 22, Yes], [22, 3, 21, 21, Yes], [22, 4, 20, 20, Yes], [22, 5, 19, 19, Yes], [22, 6, 18, 18, Yes], [22, 7, 17, 17, Yes], [22, 8, 16, 16, Yes], [22, 9, 15, 15, Yes], [22, 10, 14, 14, Yes], [22, 11, 13, 13, Yes], [22, 12, 12, 12, Yes], [22, 13, 11, 11, Yes], [22, 14, 10, 10, Yes], [22, 15, 9, 9, Yes], [22, 16, 8, 8, Yes], [22, 17, 7, 7, Yes], [22, 18, 6, 6, Yes], [22, 19, 5, 5, Yes], [22, 20, 4, 4, Yes], [22, 21, 3, 3, Yes], [22, 22, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)
List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (23, 23, [0, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 22, 22, Yes], [0, 2, 21, 21, Yes], [0, 3, 20, 20, Yes], [0, 4, 19, 19, Yes], [0, 5, 18, 18, Yes], [0, 6, 17, 17, Yes], [0, 7, 16, 16, Yes], [0, 8, 15, 15, Yes], [0, 9, 14, 14, Yes], [0, 10, 13, 13, Yes], [0, 11, 12, 12, Yes], [0, 12, 11, 11, Yes], [0, 13, 10, 10, Yes], [0, 14, 9, 9, Yes], [0, 15, 8, 8, Yes], [0, 16, 7, 7, Yes], [0, 17, 6, 6, Yes], [0, 18, 5, 5, Yes], [0, 19, 4, 4, Yes], [0, 20, 3, 3, Yes], [0, 21, 2, 2, Yes], [0, 22, 1, 1, Yes], [1, 0, 22, 22, Yes], [1, 1, 21, 21, Yes], [1, 2, 20, 20, Yes], [1, 3, 19, 19, Yes], [1, 4, 18, 18, Yes], [1, 5, 17, 17, Yes], [1, 6, 16, 16, Yes], [1, 7, 15, 15, Yes], [1, 8, 14, 14, Yes], [1, 9, 13, 13, Yes], [1, 10, 12, 12, Yes], [1, 11, 11, 11, Yes], [1, 12, 10, 10, Yes], [1, 13, 9, 9, Yes], [1, 14, 8, 8, Yes], [1, 15, 7, 7, Yes], [1, 16, 6, 6, Yes], [1, 17, 5, 5, Yes], [1, 18, 4, 4, Yes], [1, 19, 3, 3, Yes], [1, 20, 2, 2, Yes], [1, 21, 1, 1, Yes], [1, 22, 0, 0, Yes], [2, 0, 21, 21, Yes], [2, 1, 20, 20, Yes], [2, 2, 19, 19, Yes], [2, 3, 18, 18, Yes], [2, 4, 17, 17, Yes], [2, 5, 16, 16, Yes], [2, 6, 15, 15, Yes], [2, 7, 14, 14, Yes], [2, 8, 13, 13, Yes], [2, 9, 12, 12, Yes], [2, 10, 11, 11, Yes], [2, 11, 10, 10, Yes], [2, 12, 9, 9, Yes], [2, 13, 8, 8, Yes], [2, 14, 7, 7, Yes], [2, 15, 6, 6, Yes], [2, 16, 5, 5, Yes], [2, 17, 4, 4, Yes], [2, 18, 3, 3, Yes], [2, 19, 2, 2, Yes], [2, 20, 1, 1, Yes], [2, 21, 0, 0, Yes], [2, 22, 22, 22, Yes], [3, 0, 20, 20, Yes], [3, 1, 19, 19, Yes], [3, 2, 18, 18, Yes], [3, 3, 17, 17, Yes], [3, 4, 16, 16, Yes], [3, 5, 15, 15, Yes], [3, 6, 14, 14, Yes], [3, 7, 13, 13, Yes], [3, 8, 12, 12, Yes], [3, 9, 11, 11, Yes], [3, 10, 10, 10, Yes], [3, 11, 9, 9, Yes], [3, 12, 8, 8, Yes], [3, 13, 7, 7, Yes], [3, 14, 6, 6, Yes], [3, 15, 5, 5, Yes], [3, 16, 4, 4, Yes], [3, 17, 3, 3, Yes], [3, 18, 2, 2, Yes], [3, 19, 1, 1, Yes], [3, 20, 0, 0, Yes], [3, 21, 22, 22, Yes], [3, 22, 21, 21, Yes], [4, 0, 19, 19, Yes], [4, 1, 18, 18, Yes], [4, 2, 17, 17, Yes], [4, 3, 16, 16, Yes], [4, 4, 15, 15, Yes], [4, 5, 14, 14, Yes], [4, 6, 13, 13, Yes], [4, 7, 12, 12, Yes], [4, 8, 11, 11, Yes], [4, 9, 10, 10, Yes], [4, 10, 9, 9, Yes], [4, 11, 8, 8, Yes], [4, 12, 7, 7, Yes], [4, 13, 6, 6, Yes], [4, 14, 5, 5, Yes], [4, 15, 4, 4, Yes], [4, 16, 3, 3, Yes], [4, 17, 2, 2, Yes], [4, 18, 1, 1, Yes], [4, 19, 0, 0, Yes], [4, 20, 22, 22, Yes], [4, 21, 21, 21, Yes], [4, 22, 20, 20, Yes], [5, 0, 18, 18, Yes], [5, 1, 17, 17, Yes], [5, 2, 16, 16, Yes], [5, 3, 15, 15, Yes], [5, 4, 14, 14, Yes], [5, 5, 13, 13, Yes], [5, 6, 12, 12, Yes], [5, 7, 11, 11, Yes], [5, 8, 10, 10, Yes], [5, 9, 9, 9, Yes], [5, 10, 8, 8, Yes], [5, 11, 7, 7, Yes], [5, 12, 6, 6, Yes], [5, 13, 5, 5, Yes], [5, 14, 4, 4, Yes], [5, 15, 3, 3, Yes], [5, 16, 2, 2, Yes], [5, 17, 1, 1, Yes], [5, 18, 0, 0, Yes], [5, 19, 22, 22, Yes], [5, 20, 21, 21, Yes], [5, 21, 20, 20, Yes], [5, 22, 19, 19, Yes], [6, 0, 17, 17, Yes], [6, 1, 16, 16, Yes], [6, 2, 15, 15, Yes], [6, 3, 14, 14, Yes], [6, 4, 13, 13, Yes], [6, 5, 12, 12, Yes], [6, 6, 11, 11, Yes], [6, 7, 10, 10, Yes], [6, 8, 9, 9, Yes], [6, 9, 8, 8, Yes], [6, 10, 7, 7, Yes], [6, 11, 6, 6, Yes], [6, 12, 5, 5, Yes], [6, 13, 4, 4, Yes], [6, 14, 3, 3, Yes], [6, 15, 2, 2, Yes], [6, 16, 1, 1, Yes], [6, 17, 0, 0, Yes], [6, 18, 22, 22, Yes], [6, 19, 21, 21, Yes], [6, 20, 20, 20, Yes], [6, 21, 19, 19, Yes], [6, 22, 18, 18, Yes], [7, 0, 16, 16, Yes], [7, 1, 15, 15, Yes], [7, 2, 14, 14, Yes], [7, 3, 13, 13, Yes], [7, 4, 12, 12, Yes], [7, 5, 11, 11, Yes], [7, 6, 10, 10, Yes], [7, 7, 9, 9, Yes], [7, 8, 8, 8, Yes], [7, 9, 7, 7, Yes], [7, 10, 6, 6, Yes], [7, 11, 5, 5, Yes], [7, 12, 4, 4, Yes], [7, 13, 3, 3, Yes], [7, 14, 2, 2, Yes], [7, 15, 1, 1, Yes], [7, 16, 0, 0, Yes], [7, 17, 22, 22, Yes], [7, 18, 21, 21, Yes], [7, 19, 20, 20, Yes], [7, 20, 19, 19, Yes], [7, 21, 18, 18, Yes], [7, 22, 17, 17, Yes], [8, 0, 15, 15, Yes], [8, 1, 14, 14, Yes], [8, 2, 13, 13, Yes], [8, 3, 12, 12, Yes], [8, 4, 11, 11, Yes], [8, 5, 10, 10, Yes], [8, 6, 9, 9, Yes], [8, 7, 8, 8, Yes], [8, 8, 7, 7, Yes], [8, 9, 6, 6, Yes], [8, 10, 5, 5, Yes], [8, 11, 4, 4, Yes], [8, 12, 3, 3, Yes], [8, 13, 2, 2, Yes], [8, 14, 1, 1, Yes], [8, 15, 0, 0, Yes], [8, 16, 22, 22, Yes], [8, 17, 21, 21, Yes], [8, 18, 20, 20, Yes], [8, 19, 19, 19, Yes], [8, 20, 18, 18, Yes], [8, 21, 17, 17, Yes], [8, 22, 16, 16, Yes], [9, 0, 14, 14, Yes], [9, 1, 13, 13, Yes], [9, 2, 12, 12, Yes], [9, 3, 11, 11, Yes], [9, 4, 10, 10, Yes], [9, 5, 9, 9, Yes], [9, 6, 8, 8, Yes], [9, 7, 7, 7, Yes], [9, 8, 6, 6, Yes], [9, 9, 5, 5, Yes], [9, 10, 4, 4, Yes], [9, 11, 3, 3, Yes], [9, 12, 2, 2, Yes], [9, 13, 1, 1, Yes], [9, 14, 0, 0, Yes], [9, 15, 22, 22, Yes], [9, 16, 21, 21, Yes], [9, 17, 20, 20, Yes], [9, 18, 19, 19, Yes], [9, 19, 18, 18, Yes], [9, 20, 17, 17, Yes], [9, 21, 16, 16, Yes], [9, 22, 15, 15, Yes], [10, 0, 13, 13, Yes], [10, 1, 12, 12, Yes], [10, 2, 11, 11, Yes], [10, 3, 10, 10, Yes], [10, 4, 9, 9, Yes], [10, 5, 8, 8, Yes], [10, 6, 7, 7, Yes], [10, 7, 6, 6, Yes], [10, 8, 5, 5, Yes], [10, 9, 4, 4, Yes], [10, 10, 3, 3, Yes], [10, 11, 2, 2, Yes], [10, 12, 1, 1, Yes], [10, 13, 0, 0, Yes], [10, 14, 22, 22, Yes], [10, 15, 21, 21, Yes], [10, 16, 20, 20, Yes], [10, 17, 19, 19, Yes], [10, 18, 18, 18, Yes], [10, 19, 17, 17, Yes], [10, 20, 16, 16, Yes], [10, 21, 15, 15, Yes], [10, 22, 14, 14, Yes], [11, 0, 12, 12, Yes], [11, 1, 11, 11, Yes], [11, 2, 10, 10, Yes], [11, 3, 9, 9, Yes], [11, 4, 8, 8, Yes], [11, 5, 7, 7, Yes], [11, 6, 6, 6, Yes], [11, 7, 5, 5, Yes], [11, 8, 4, 4, Yes], [11, 9, 3, 3, Yes], [11, 10, 2, 2, Yes], [11, 11, 1, 1, Yes], [11, 12, 0, 0, Yes], [11, 13, 22, 22, Yes], [11, 14, 21, 21, Yes], [11, 15, 20, 20, Yes], [11, 16, 19, 19, Yes], [11, 17, 18, 18, Yes], [11, 18, 17, 17, Yes], [11, 19, 16, 16, Yes], [11, 20, 15, 15, Yes], [11, 21, 14, 14, Yes], [11, 22, 13, 13, Yes], [12, 0, 11, 11, Yes], [12, 1, 10, 10, Yes], [12, 2, 9, 9, Yes], [12, 3, 8, 8, Yes], [12, 4, 7, 7, Yes], [12, 5, 6, 6, Yes], [12, 6, 5, 5, Yes], [12, 7, 4, 4, Yes], [12, 8, 3, 3, Yes], [12, 9, 2, 2, Yes], [12, 10, 1, 1, Yes], [12, 11, 0, 0, Yes], [12, 12, 22, 22, Yes], [12, 13, 21, 21, Yes], [12, 14, 20, 20, Yes], [12, 15, 19, 19, Yes], [12, 16, 18, 18, Yes], [12, 17, 17, 17, Yes], [12, 18, 16, 16, Yes], [12, 19, 15, 15, Yes], [12, 20, 14, 14, Yes], [12, 21, 13, 13, Yes], [12, 22, 12, 12, Yes], [13, 0, 10, 10, Yes], [13, 1, 9, 9, Yes], [13, 2, 8, 8, Yes], [13, 3, 7, 7, Yes], [13, 4, 6, 6, Yes], [13, 5, 5, 5, Yes], [13, 6, 4, 4, Yes], [13, 7, 3, 3, Yes], [13, 8, 2, 2, Yes], [13, 9, 1, 1, Yes], [13, 10, 0, 0, Yes], [13, 11, 22, 22, Yes], [13, 12, 21, 21, Yes], [13, 13, 20, 20, Yes], [13, 14, 19, 19, Yes], [13, 15, 18, 18, Yes], [13, 16, 17, 17, Yes], [13, 17, 16, 16, Yes], [13, 18, 15, 15, Yes], [13, 19, 14, 14, Yes], [13, 20, 13, 13, Yes], [13, 21, 12, 12, Yes], [13, 22, 11, 11, Yes], [14, 0, 9, 9, Yes], [14, 1, 8, 8, Yes], [14, 2, 7, 7, Yes], [14, 3, 6, 6, Yes], [14, 4, 5, 5, Yes], [14, 5, 4, 4, Yes], [14, 6, 3, 3, Yes], [14, 7, 2, 2, Yes], [14, 8, 1, 1, Yes], [14, 9, 0, 0, Yes], [14, 10, 22, 22, Yes], [14, 11, 21, 21, Yes], [14, 12, 20, 20, Yes], [14, 13, 19, 19, Yes], [14, 14, 18, 18, Yes], [14, 15, 17, 17, Yes], [14, 16, 16, 16, Yes], [14, 17, 15, 15, Yes], [14, 18, 14, 14, Yes], [14, 19, 13, 13, Yes], [14, 20, 12, 12, Yes], [14, 21, 11, 11, Yes], [14, 22, 10, 10, Yes], [15, 0, 8, 8, Yes], [15, 1, 7, 7, Yes], [15, 2, 6, 6, Yes], [15, 3, 5, 5, Yes], [15, 4, 4, 4, Yes], [15, 5, 3, 3, Yes], [15, 6, 2, 2, Yes], [15, 7, 1, 1, Yes], [15, 8, 0, 0, Yes], [15, 9, 22, 22, Yes], [15, 10, 21, 21, Yes], [15, 11, 20, 20, Yes], [15, 12, 19, 19, Yes], [15, 13, 18, 18, Yes], [15, 14, 17, 17, Yes], [15, 15, 16, 16, Yes], [15, 16, 15, 15, Yes], [15, 17, 14, 14, Yes], [15, 18, 13, 13, Yes], [15, 19, 12, 12, Yes], [15, 20, 11, 11, Yes], [15, 21, 10, 10, Yes], [15, 22, 9, 9, Yes], [16, 0, 7, 7, Yes], [16, 1, 6, 6, Yes], [16, 2, 5, 5, Yes], [16, 3, 4, 4, Yes], [16, 4, 3, 3, Yes], [16, 5, 2, 2, Yes], [16, 6, 1, 1, Yes], [16, 7, 0, 0, Yes], [16, 8, 22, 22, Yes], [16, 9, 21, 21, Yes], [16, 10, 20, 20, Yes], [16, 11, 19, 19, Yes], [16, 12, 18, 18, Yes], [16, 13, 17, 17, Yes], [16, 14, 16, 16, Yes], [16, 15, 15, 15, Yes], [16, 16, 14, 14, Yes], [16, 17, 13, 13, Yes], [16, 18, 12, 12, Yes], [16, 19, 11, 11, Yes], [16, 20, 10, 10, Yes], [16, 21, 9, 9, Yes], [16, 22, 8, 8, Yes], [17, 0, 6, 6, Yes], [17, 1, 5, 5, Yes], [17, 2, 4, 4, Yes], [17, 3, 3, 3, Yes], [17, 4, 2, 2, Yes], [17, 5, 1, 1, Yes], [17, 6, 0, 0, Yes], [17, 7, 22, 22, Yes], [17, 8, 21, 21, Yes], [17, 9, 20, 20, Yes], [17, 10, 19, 19, Yes], [17, 11, 18, 18, Yes], [17, 12, 17, 17, Yes], [17, 13, 16, 16, Yes], [17, 14, 15, 15, Yes], [17, 15, 14, 14, Yes], [17, 16, 13, 13, Yes], [17, 17, 12, 12, Yes], [17, 18, 11, 11, Yes], [17, 19, 10, 10, Yes], [17, 20, 9, 9, Yes], [17, 21, 8, 8, Yes], [17, 22, 7, 7, Yes], [18, 0, 5, 5, Yes], [18, 1, 4, 4, Yes], [18, 2, 3, 3, Yes], [18, 3, 2, 2, Yes], [18, 4, 1, 1, Yes], [18, 5, 0, 0, Yes], [18, 6, 22, 22, Yes], [18, 7, 21, 21, Yes], [18, 8, 20, 20, Yes], [18, 9, 19, 19, Yes], [18, 10, 18, 18, Yes], [18, 11, 17, 17, Yes], [18, 12, 16, 16, Yes], [18, 13, 15, 15, Yes], [18, 14, 14, 14, Yes], [18, 15, 13, 13, Yes], [18, 16, 12, 12, Yes], [18, 17, 11, 11, Yes], [18, 18, 10, 10, Yes], [18, 19, 9, 9, Yes], [18, 20, 8, 8, Yes], [18, 21, 7, 7, Yes], [18, 22, 6, 6, Yes], [19, 0, 4, 4, Yes], [19, 1, 3, 3, Yes], [19, 2, 2, 2, Yes], [19, 3, 1, 1, Yes], [19, 4, 0, 0, Yes], [19, 5, 22, 22, Yes], [19, 6, 21, 21, Yes], [19, 7, 20, 20, Yes], [19, 8, 19, 19, Yes], [19, 9, 18, 18, Yes], [19, 10, 17, 17, Yes], [19, 11, 16, 16, Yes], [19, 12, 15, 15, Yes], [19, 13, 14, 14, Yes], [19, 14, 13, 13, Yes], [19, 15, 12, 12, Yes], [19, 16, 11, 11, Yes], [19, 17, 10, 10, Yes], [19, 18, 9, 9, Yes], [19, 19, 8, 8, Yes], [19, 20, 7, 7, Yes], [19, 21, 6, 6, Yes], [19, 22, 5, 5, Yes], [20, 0, 3, 3, Yes], [20, 1, 2, 2, Yes], [20, 2, 1, 1, Yes], [20, 3, 0, 0, Yes], [20, 4, 22, 22, Yes], [20, 5, 21, 21, Yes], [20, 6, 20, 20, Yes], [20, 7, 19, 19, Yes], [20, 8, 18, 18, Yes], [20, 9, 17, 17, Yes], [20, 10, 16, 16, Yes], [20, 11, 15, 15, Yes], [20, 12, 14, 14, Yes], [20, 13, 13, 13, Yes], [20, 14, 12, 12, Yes], [20, 15, 11, 11, Yes], [20, 16, 10, 10, Yes], [20, 17, 9, 9, Yes], [20, 18, 8, 8, Yes], [20, 19, 7, 7, Yes], [20, 20, 6, 6, Yes], [20, 21, 5, 5, Yes], [20, 22, 4, 4, Yes], [21, 0, 2, 2, Yes], [21, 1, 1, 1, Yes], [21, 2, 0, 0, Yes], [21, 3, 22, 22, Yes], [21, 4, 21, 21, Yes], [21, 5, 20, 20, Yes], [21, 6, 19, 19, Yes], [21, 7, 18, 18, Yes], [21, 8, 17, 17, Yes], [21, 9, 16, 16, Yes], [21, 10, 15, 15, Yes], [21, 11, 14, 14, Yes], [21, 12, 13, 13, Yes], [21, 13, 12, 12, Yes], [21, 14, 11, 11, Yes], [21, 15, 10, 10, Yes], [21, 16, 9, 9, Yes], [21, 17, 8, 8, Yes], [21, 18, 7, 7, Yes], [21, 19, 6, 6, Yes], [21, 20, 5, 5, Yes], [21, 21, 4, 4, Yes], [21, 22, 3, 3, Yes], [22, 0, 1, 1, Yes], [22, 1, 0, 0, Yes], [22, 2, 22, 22, Yes], [22, 3, 21, 21, Yes], [22, 4, 20, 20, Yes], [22, 5, 19, 19, Yes], [22, 6, 18, 18, Yes], [22, 7, 17, 17, Yes], [22, 8, 16, 16, Yes], [22, 9, 15, 15, Yes], [22, 10, 14, 14, Yes], [22, 11, 13, 13, Yes], [22, 12, 12, 12, Yes], [22, 13, 11, 11, Yes], [22, 14, 10, 10, Yes], [22, 15, 9, 9, Yes], [22, 16, 8, 8, Yes], [22, 17, 7, 7, Yes], [22, 18, 6, 6, Yes], [22, 19, 5, 5, Yes], [22, 20, 4, 4, Yes], [22, 21, 3, 3, Yes], [22, 22, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-homomorphism-qualification · candidate
22. Qualify Both Groups and Every Product Pair before Applying Kernel Theorems
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]], target identity=0, and total mapping=[0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).
View answer and explanation
(24, 24, [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 23, 23, Yes], [0, 2, 22, 22, Yes], [0, 3, 21, 21, Yes], [0, 4, 20, 20, Yes], [0, 5, 19, 19, Yes], [0, 6, 18, 18, Yes], [0, 7, 17, 17, Yes], [0, 8, 16, 16, Yes], [0, 9, 15, 15, Yes], [0, 10, 14, 14, Yes], [0, 11, 13, 13, Yes], [0, 12, 12, 12, Yes], [0, 13, 11, 11, Yes], [0, 14, 10, 10, Yes], [0, 15, 9, 9, Yes], [0, 16, 8, 8, Yes], [0, 17, 7, 7, Yes], [0, 18, 6, 6, Yes], [0, 19, 5, 5, Yes], [0, 20, 4, 4, Yes], [0, 21, 3, 3, Yes], [0, 22, 2, 2, Yes], [0, 23, 1, 1, Yes], [1, 0, 23, 23, Yes], [1, 1, 22, 22, Yes], [1, 2, 21, 21, Yes], [1, 3, 20, 20, Yes], [1, 4, 19, 19, Yes], [1, 5, 18, 18, Yes], [1, 6, 17, 17, Yes], [1, 7, 16, 16, Yes], [1, 8, 15, 15, Yes], [1, 9, 14, 14, Yes], [1, 10, 13, 13, Yes], [1, 11, 12, 12, Yes], [1, 12, 11, 11, Yes], [1, 13, 10, 10, Yes], [1, 14, 9, 9, Yes], [1, 15, 8, 8, Yes], [1, 16, 7, 7, Yes], [1, 17, 6, 6, Yes], [1, 18, 5, 5, Yes], [1, 19, 4, 4, Yes], [1, 20, 3, 3, Yes], [1, 21, 2, 2, Yes], [1, 22, 1, 1, Yes], [1, 23, 0, 0, Yes], [2, 0, 22, 22, Yes], [2, 1, 21, 21, Yes], [2, 2, 20, 20, Yes], [2, 3, 19, 19, Yes], [2, 4, 18, 18, Yes], [2, 5, 17, 17, Yes], [2, 6, 16, 16, Yes], [2, 7, 15, 15, Yes], [2, 8, 14, 14, Yes], [2, 9, 13, 13, Yes], [2, 10, 12, 12, Yes], [2, 11, 11, 11, Yes], [2, 12, 10, 10, Yes], [2, 13, 9, 9, Yes], [2, 14, 8, 8, Yes], [2, 15, 7, 7, Yes], [2, 16, 6, 6, Yes], [2, 17, 5, 5, Yes], [2, 18, 4, 4, Yes], [2, 19, 3, 3, Yes], [2, 20, 2, 2, Yes], [2, 21, 1, 1, Yes], [2, 22, 0, 0, Yes], [2, 23, 23, 23, Yes], [3, 0, 21, 21, Yes], [3, 1, 20, 20, Yes], [3, 2, 19, 19, Yes], [3, 3, 18, 18, Yes], [3, 4, 17, 17, Yes], [3, 5, 16, 16, Yes], [3, 6, 15, 15, Yes], [3, 7, 14, 14, Yes], [3, 8, 13, 13, Yes], [3, 9, 12, 12, Yes], [3, 10, 11, 11, Yes], [3, 11, 10, 10, Yes], [3, 12, 9, 9, Yes], [3, 13, 8, 8, Yes], [3, 14, 7, 7, Yes], [3, 15, 6, 6, Yes], [3, 16, 5, 5, Yes], [3, 17, 4, 4, Yes], [3, 18, 3, 3, Yes], [3, 19, 2, 2, Yes], [3, 20, 1, 1, Yes], [3, 21, 0, 0, Yes], [3, 22, 23, 23, Yes], [3, 23, 22, 22, Yes], [4, 0, 20, 20, Yes], [4, 1, 19, 19, Yes], [4, 2, 18, 18, Yes], [4, 3, 17, 17, Yes], [4, 4, 16, 16, Yes], [4, 5, 15, 15, Yes], [4, 6, 14, 14, Yes], [4, 7, 13, 13, Yes], [4, 8, 12, 12, Yes], [4, 9, 11, 11, Yes], [4, 10, 10, 10, Yes], [4, 11, 9, 9, Yes], [4, 12, 8, 8, Yes], [4, 13, 7, 7, Yes], [4, 14, 6, 6, Yes], [4, 15, 5, 5, Yes], [4, 16, 4, 4, Yes], [4, 17, 3, 3, Yes], [4, 18, 2, 2, Yes], [4, 19, 1, 1, Yes], [4, 20, 0, 0, Yes], [4, 21, 23, 23, Yes], [4, 22, 22, 22, Yes], [4, 23, 21, 21, Yes], [5, 0, 19, 19, Yes], [5, 1, 18, 18, Yes], [5, 2, 17, 17, Yes], [5, 3, 16, 16, Yes], [5, 4, 15, 15, Yes], [5, 5, 14, 14, Yes], [5, 6, 13, 13, Yes], [5, 7, 12, 12, Yes], [5, 8, 11, 11, Yes], [5, 9, 10, 10, Yes], [5, 10, 9, 9, Yes], [5, 11, 8, 8, Yes], [5, 12, 7, 7, Yes], [5, 13, 6, 6, Yes], [5, 14, 5, 5, Yes], [5, 15, 4, 4, Yes], [5, 16, 3, 3, Yes], [5, 17, 2, 2, Yes], [5, 18, 1, 1, Yes], [5, 19, 0, 0, Yes], [5, 20, 23, 23, Yes], [5, 21, 22, 22, Yes], [5, 22, 21, 21, Yes], [5, 23, 20, 20, Yes], [6, 0, 18, 18, Yes], [6, 1, 17, 17, Yes], [6, 2, 16, 16, Yes], [6, 3, 15, 15, Yes], [6, 4, 14, 14, Yes], [6, 5, 13, 13, Yes], [6, 6, 12, 12, Yes], [6, 7, 11, 11, Yes], [6, 8, 10, 10, Yes], [6, 9, 9, 9, Yes], [6, 10, 8, 8, Yes], [6, 11, 7, 7, Yes], [6, 12, 6, 6, Yes], [6, 13, 5, 5, Yes], [6, 14, 4, 4, Yes], [6, 15, 3, 3, Yes], [6, 16, 2, 2, Yes], [6, 17, 1, 1, Yes], [6, 18, 0, 0, Yes], [6, 19, 23, 23, Yes], [6, 20, 22, 22, Yes], [6, 21, 21, 21, Yes], [6, 22, 20, 20, Yes], [6, 23, 19, 19, Yes], [7, 0, 17, 17, Yes], [7, 1, 16, 16, Yes], [7, 2, 15, 15, Yes], [7, 3, 14, 14, Yes], [7, 4, 13, 13, Yes], [7, 5, 12, 12, Yes], [7, 6, 11, 11, Yes], [7, 7, 10, 10, Yes], [7, 8, 9, 9, Yes], [7, 9, 8, 8, Yes], [7, 10, 7, 7, Yes], [7, 11, 6, 6, Yes], [7, 12, 5, 5, Yes], [7, 13, 4, 4, Yes], [7, 14, 3, 3, Yes], [7, 15, 2, 2, Yes], [7, 16, 1, 1, Yes], [7, 17, 0, 0, Yes], [7, 18, 23, 23, Yes], [7, 19, 22, 22, Yes], [7, 20, 21, 21, Yes], [7, 21, 20, 20, Yes], [7, 22, 19, 19, Yes], [7, 23, 18, 18, Yes], [8, 0, 16, 16, Yes], [8, 1, 15, 15, Yes], [8, 2, 14, 14, Yes], [8, 3, 13, 13, Yes], [8, 4, 12, 12, Yes], [8, 5, 11, 11, Yes], [8, 6, 10, 10, Yes], [8, 7, 9, 9, Yes], [8, 8, 8, 8, Yes], [8, 9, 7, 7, Yes], [8, 10, 6, 6, Yes], [8, 11, 5, 5, Yes], [8, 12, 4, 4, Yes], [8, 13, 3, 3, Yes], [8, 14, 2, 2, Yes], [8, 15, 1, 1, Yes], [8, 16, 0, 0, Yes], [8, 17, 23, 23, Yes], [8, 18, 22, 22, Yes], [8, 19, 21, 21, Yes], [8, 20, 20, 20, Yes], [8, 21, 19, 19, Yes], [8, 22, 18, 18, Yes], [8, 23, 17, 17, Yes], [9, 0, 15, 15, Yes], [9, 1, 14, 14, Yes], [9, 2, 13, 13, Yes], [9, 3, 12, 12, Yes], [9, 4, 11, 11, Yes], [9, 5, 10, 10, Yes], [9, 6, 9, 9, Yes], [9, 7, 8, 8, Yes], [9, 8, 7, 7, Yes], [9, 9, 6, 6, Yes], [9, 10, 5, 5, Yes], [9, 11, 4, 4, Yes], [9, 12, 3, 3, Yes], [9, 13, 2, 2, Yes], [9, 14, 1, 1, Yes], [9, 15, 0, 0, Yes], [9, 16, 23, 23, Yes], [9, 17, 22, 22, Yes], [9, 18, 21, 21, Yes], [9, 19, 20, 20, Yes], [9, 20, 19, 19, Yes], [9, 21, 18, 18, Yes], [9, 22, 17, 17, Yes], [9, 23, 16, 16, Yes], [10, 0, 14, 14, Yes], [10, 1, 13, 13, Yes], [10, 2, 12, 12, Yes], [10, 3, 11, 11, Yes], [10, 4, 10, 10, Yes], [10, 5, 9, 9, Yes], [10, 6, 8, 8, Yes], [10, 7, 7, 7, Yes], [10, 8, 6, 6, Yes], [10, 9, 5, 5, Yes], [10, 10, 4, 4, Yes], [10, 11, 3, 3, Yes], [10, 12, 2, 2, Yes], [10, 13, 1, 1, Yes], [10, 14, 0, 0, Yes], [10, 15, 23, 23, Yes], [10, 16, 22, 22, Yes], [10, 17, 21, 21, Yes], [10, 18, 20, 20, Yes], [10, 19, 19, 19, Yes], [10, 20, 18, 18, Yes], [10, 21, 17, 17, Yes], [10, 22, 16, 16, Yes], [10, 23, 15, 15, Yes], [11, 0, 13, 13, Yes], [11, 1, 12, 12, Yes], [11, 2, 11, 11, Yes], [11, 3, 10, 10, Yes], [11, 4, 9, 9, Yes], [11, 5, 8, 8, Yes], [11, 6, 7, 7, Yes], [11, 7, 6, 6, Yes], [11, 8, 5, 5, Yes], [11, 9, 4, 4, Yes], [11, 10, 3, 3, Yes], [11, 11, 2, 2, Yes], [11, 12, 1, 1, Yes], [11, 13, 0, 0, Yes], [11, 14, 23, 23, Yes], [11, 15, 22, 22, Yes], [11, 16, 21, 21, Yes], [11, 17, 20, 20, Yes], [11, 18, 19, 19, Yes], [11, 19, 18, 18, Yes], [11, 20, 17, 17, Yes], [11, 21, 16, 16, Yes], [11, 22, 15, 15, Yes], [11, 23, 14, 14, Yes], [12, 0, 12, 12, Yes], [12, 1, 11, 11, Yes], [12, 2, 10, 10, Yes], [12, 3, 9, 9, Yes], [12, 4, 8, 8, Yes], [12, 5, 7, 7, Yes], [12, 6, 6, 6, Yes], [12, 7, 5, 5, Yes], [12, 8, 4, 4, Yes], [12, 9, 3, 3, Yes], [12, 10, 2, 2, Yes], [12, 11, 1, 1, Yes], [12, 12, 0, 0, Yes], [12, 13, 23, 23, Yes], [12, 14, 22, 22, Yes], [12, 15, 21, 21, Yes], [12, 16, 20, 20, Yes], [12, 17, 19, 19, Yes], [12, 18, 18, 18, Yes], [12, 19, 17, 17, Yes], [12, 20, 16, 16, Yes], [12, 21, 15, 15, Yes], [12, 22, 14, 14, Yes], [12, 23, 13, 13, Yes], [13, 0, 11, 11, Yes], [13, 1, 10, 10, Yes], [13, 2, 9, 9, Yes], [13, 3, 8, 8, Yes], [13, 4, 7, 7, Yes], [13, 5, 6, 6, Yes], [13, 6, 5, 5, Yes], [13, 7, 4, 4, Yes], [13, 8, 3, 3, Yes], [13, 9, 2, 2, Yes], [13, 10, 1, 1, Yes], [13, 11, 0, 0, Yes], [13, 12, 23, 23, Yes], [13, 13, 22, 22, Yes], [13, 14, 21, 21, Yes], [13, 15, 20, 20, Yes], [13, 16, 19, 19, Yes], [13, 17, 18, 18, Yes], [13, 18, 17, 17, Yes], [13, 19, 16, 16, Yes], [13, 20, 15, 15, Yes], [13, 21, 14, 14, Yes], [13, 22, 13, 13, Yes], [13, 23, 12, 12, Yes], [14, 0, 10, 10, Yes], [14, 1, 9, 9, Yes], [14, 2, 8, 8, Yes], [14, 3, 7, 7, Yes], [14, 4, 6, 6, Yes], [14, 5, 5, 5, Yes], [14, 6, 4, 4, Yes], [14, 7, 3, 3, Yes], [14, 8, 2, 2, Yes], [14, 9, 1, 1, Yes], [14, 10, 0, 0, Yes], [14, 11, 23, 23, Yes], [14, 12, 22, 22, Yes], [14, 13, 21, 21, Yes], [14, 14, 20, 20, Yes], [14, 15, 19, 19, Yes], [14, 16, 18, 18, Yes], [14, 17, 17, 17, Yes], [14, 18, 16, 16, Yes], [14, 19, 15, 15, Yes], [14, 20, 14, 14, Yes], [14, 21, 13, 13, Yes], [14, 22, 12, 12, Yes], [14, 23, 11, 11, Yes], [15, 0, 9, 9, Yes], [15, 1, 8, 8, Yes], [15, 2, 7, 7, Yes], [15, 3, 6, 6, Yes], [15, 4, 5, 5, Yes], [15, 5, 4, 4, Yes], [15, 6, 3, 3, Yes], [15, 7, 2, 2, Yes], [15, 8, 1, 1, Yes], [15, 9, 0, 0, Yes], [15, 10, 23, 23, Yes], [15, 11, 22, 22, Yes], [15, 12, 21, 21, Yes], [15, 13, 20, 20, Yes], [15, 14, 19, 19, Yes], [15, 15, 18, 18, Yes], [15, 16, 17, 17, Yes], [15, 17, 16, 16, Yes], [15, 18, 15, 15, Yes], [15, 19, 14, 14, Yes], [15, 20, 13, 13, Yes], [15, 21, 12, 12, Yes], [15, 22, 11, 11, Yes], [15, 23, 10, 10, Yes], [16, 0, 8, 8, Yes], [16, 1, 7, 7, Yes], [16, 2, 6, 6, Yes], [16, 3, 5, 5, Yes], [16, 4, 4, 4, Yes], [16, 5, 3, 3, Yes], [16, 6, 2, 2, Yes], [16, 7, 1, 1, Yes], [16, 8, 0, 0, Yes], [16, 9, 23, 23, Yes], [16, 10, 22, 22, Yes], [16, 11, 21, 21, Yes], [16, 12, 20, 20, Yes], [16, 13, 19, 19, Yes], [16, 14, 18, 18, Yes], [16, 15, 17, 17, Yes], [16, 16, 16, 16, Yes], [16, 17, 15, 15, Yes], [16, 18, 14, 14, Yes], [16, 19, 13, 13, Yes], [16, 20, 12, 12, Yes], [16, 21, 11, 11, Yes], [16, 22, 10, 10, Yes], [16, 23, 9, 9, Yes], [17, 0, 7, 7, Yes], [17, 1, 6, 6, Yes], [17, 2, 5, 5, Yes], [17, 3, 4, 4, Yes], [17, 4, 3, 3, Yes], [17, 5, 2, 2, Yes], [17, 6, 1, 1, Yes], [17, 7, 0, 0, Yes], [17, 8, 23, 23, Yes], [17, 9, 22, 22, Yes], [17, 10, 21, 21, Yes], [17, 11, 20, 20, Yes], [17, 12, 19, 19, Yes], [17, 13, 18, 18, Yes], [17, 14, 17, 17, Yes], [17, 15, 16, 16, Yes], [17, 16, 15, 15, Yes], [17, 17, 14, 14, Yes], [17, 18, 13, 13, Yes], [17, 19, 12, 12, Yes], [17, 20, 11, 11, Yes], [17, 21, 10, 10, Yes], [17, 22, 9, 9, Yes], [17, 23, 8, 8, Yes], [18, 0, 6, 6, Yes], [18, 1, 5, 5, Yes], [18, 2, 4, 4, Yes], [18, 3, 3, 3, Yes], [18, 4, 2, 2, Yes], [18, 5, 1, 1, Yes], [18, 6, 0, 0, Yes], [18, 7, 23, 23, Yes], [18, 8, 22, 22, Yes], [18, 9, 21, 21, Yes], [18, 10, 20, 20, Yes], [18, 11, 19, 19, Yes], [18, 12, 18, 18, Yes], [18, 13, 17, 17, Yes], [18, 14, 16, 16, Yes], [18, 15, 15, 15, Yes], [18, 16, 14, 14, Yes], [18, 17, 13, 13, Yes], [18, 18, 12, 12, Yes], [18, 19, 11, 11, Yes], [18, 20, 10, 10, Yes], [18, 21, 9, 9, Yes], [18, 22, 8, 8, Yes], [18, 23, 7, 7, Yes], [19, 0, 5, 5, Yes], [19, 1, 4, 4, Yes], [19, 2, 3, 3, Yes], [19, 3, 2, 2, Yes], [19, 4, 1, 1, Yes], [19, 5, 0, 0, Yes], [19, 6, 23, 23, Yes], [19, 7, 22, 22, Yes], [19, 8, 21, 21, Yes], [19, 9, 20, 20, Yes], [19, 10, 19, 19, Yes], [19, 11, 18, 18, Yes], [19, 12, 17, 17, Yes], [19, 13, 16, 16, Yes], [19, 14, 15, 15, Yes], [19, 15, 14, 14, Yes], [19, 16, 13, 13, Yes], [19, 17, 12, 12, Yes], [19, 18, 11, 11, Yes], [19, 19, 10, 10, Yes], [19, 20, 9, 9, Yes], [19, 21, 8, 8, Yes], [19, 22, 7, 7, Yes], [19, 23, 6, 6, Yes], [20, 0, 4, 4, Yes], [20, 1, 3, 3, Yes], [20, 2, 2, 2, Yes], [20, 3, 1, 1, Yes], [20, 4, 0, 0, Yes], [20, 5, 23, 23, Yes], [20, 6, 22, 22, Yes], [20, 7, 21, 21, Yes], [20, 8, 20, 20, Yes], [20, 9, 19, 19, Yes], [20, 10, 18, 18, Yes], [20, 11, 17, 17, Yes], [20, 12, 16, 16, Yes], [20, 13, 15, 15, Yes], [20, 14, 14, 14, Yes], [20, 15, 13, 13, Yes], [20, 16, 12, 12, Yes], [20, 17, 11, 11, Yes], [20, 18, 10, 10, Yes], [20, 19, 9, 9, Yes], [20, 20, 8, 8, Yes], [20, 21, 7, 7, Yes], [20, 22, 6, 6, Yes], [20, 23, 5, 5, Yes], [21, 0, 3, 3, Yes], [21, 1, 2, 2, Yes], [21, 2, 1, 1, Yes], [21, 3, 0, 0, Yes], [21, 4, 23, 23, Yes], [21, 5, 22, 22, Yes], [21, 6, 21, 21, Yes], [21, 7, 20, 20, Yes], [21, 8, 19, 19, Yes], [21, 9, 18, 18, Yes], [21, 10, 17, 17, Yes], [21, 11, 16, 16, Yes], [21, 12, 15, 15, Yes], [21, 13, 14, 14, Yes], [21, 14, 13, 13, Yes], [21, 15, 12, 12, Yes], [21, 16, 11, 11, Yes], [21, 17, 10, 10, Yes], [21, 18, 9, 9, Yes], [21, 19, 8, 8, Yes], [21, 20, 7, 7, Yes], [21, 21, 6, 6, Yes], [21, 22, 5, 5, Yes], [21, 23, 4, 4, Yes], [22, 0, 2, 2, Yes], [22, 1, 1, 1, Yes], [22, 2, 0, 0, Yes], [22, 3, 23, 23, Yes], [22, 4, 22, 22, Yes], [22, 5, 21, 21, Yes], [22, 6, 20, 20, Yes], [22, 7, 19, 19, Yes], [22, 8, 18, 18, Yes], [22, 9, 17, 17, Yes], [22, 10, 16, 16, Yes], [22, 11, 15, 15, Yes], [22, 12, 14, 14, Yes], [22, 13, 13, 13, Yes], [22, 14, 12, 12, Yes], [22, 15, 11, 11, Yes], [22, 16, 10, 10, Yes], [22, 17, 9, 9, Yes], [22, 18, 8, 8, Yes], [22, 19, 7, 7, Yes], [22, 20, 6, 6, Yes], [22, 21, 5, 5, Yes], [22, 22, 4, 4, Yes], [22, 23, 3, 3, Yes], [23, 0, 1, 1, Yes], [23, 1, 0, 0, Yes], [23, 2, 23, 23, Yes], [23, 3, 22, 22, Yes], [23, 4, 21, 21, Yes], [23, 5, 20, 20, Yes], [23, 6, 19, 19, Yes], [23, 7, 18, 18, Yes], [23, 8, 17, 17, Yes], [23, 9, 16, 16, Yes], [23, 10, 15, 15, Yes], [23, 11, 14, 14, Yes], [23, 12, 13, 13, Yes], [23, 13, 12, 12, Yes], [23, 14, 11, 11, Yes], [23, 15, 10, 10, Yes], [23, 16, 9, 9, Yes], [23, 17, 8, 8, Yes], [23, 18, 7, 7, Yes], [23, 19, 6, 6, Yes], [23, 20, 5, 5, Yes], [23, 21, 4, 4, Yes], [23, 22, 3, 3, Yes], [23, 23, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)
List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (24, 24, [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 23, 23, Yes], [0, 2, 22, 22, Yes], [0, 3, 21, 21, Yes], [0, 4, 20, 20, Yes], [0, 5, 19, 19, Yes], [0, 6, 18, 18, Yes], [0, 7, 17, 17, Yes], [0, 8, 16, 16, Yes], [0, 9, 15, 15, Yes], [0, 10, 14, 14, Yes], [0, 11, 13, 13, Yes], [0, 12, 12, 12, Yes], [0, 13, 11, 11, Yes], [0, 14, 10, 10, Yes], [0, 15, 9, 9, Yes], [0, 16, 8, 8, Yes], [0, 17, 7, 7, Yes], [0, 18, 6, 6, Yes], [0, 19, 5, 5, Yes], [0, 20, 4, 4, Yes], [0, 21, 3, 3, Yes], [0, 22, 2, 2, Yes], [0, 23, 1, 1, Yes], [1, 0, 23, 23, Yes], [1, 1, 22, 22, Yes], [1, 2, 21, 21, Yes], [1, 3, 20, 20, Yes], [1, 4, 19, 19, Yes], [1, 5, 18, 18, Yes], [1, 6, 17, 17, Yes], [1, 7, 16, 16, Yes], [1, 8, 15, 15, Yes], [1, 9, 14, 14, Yes], [1, 10, 13, 13, Yes], [1, 11, 12, 12, Yes], [1, 12, 11, 11, Yes], [1, 13, 10, 10, Yes], [1, 14, 9, 9, Yes], [1, 15, 8, 8, Yes], [1, 16, 7, 7, Yes], [1, 17, 6, 6, Yes], [1, 18, 5, 5, Yes], [1, 19, 4, 4, Yes], [1, 20, 3, 3, Yes], [1, 21, 2, 2, Yes], [1, 22, 1, 1, Yes], [1, 23, 0, 0, Yes], [2, 0, 22, 22, Yes], [2, 1, 21, 21, Yes], [2, 2, 20, 20, Yes], [2, 3, 19, 19, Yes], [2, 4, 18, 18, Yes], [2, 5, 17, 17, Yes], [2, 6, 16, 16, Yes], [2, 7, 15, 15, Yes], [2, 8, 14, 14, Yes], [2, 9, 13, 13, Yes], [2, 10, 12, 12, Yes], [2, 11, 11, 11, Yes], [2, 12, 10, 10, Yes], [2, 13, 9, 9, Yes], [2, 14, 8, 8, Yes], [2, 15, 7, 7, Yes], [2, 16, 6, 6, Yes], [2, 17, 5, 5, Yes], [2, 18, 4, 4, Yes], [2, 19, 3, 3, Yes], [2, 20, 2, 2, Yes], [2, 21, 1, 1, Yes], [2, 22, 0, 0, Yes], [2, 23, 23, 23, Yes], [3, 0, 21, 21, Yes], [3, 1, 20, 20, Yes], [3, 2, 19, 19, Yes], [3, 3, 18, 18, Yes], [3, 4, 17, 17, Yes], [3, 5, 16, 16, Yes], [3, 6, 15, 15, Yes], [3, 7, 14, 14, Yes], [3, 8, 13, 13, Yes], [3, 9, 12, 12, Yes], [3, 10, 11, 11, Yes], [3, 11, 10, 10, Yes], [3, 12, 9, 9, Yes], [3, 13, 8, 8, Yes], [3, 14, 7, 7, Yes], [3, 15, 6, 6, Yes], [3, 16, 5, 5, Yes], [3, 17, 4, 4, Yes], [3, 18, 3, 3, Yes], [3, 19, 2, 2, Yes], [3, 20, 1, 1, Yes], [3, 21, 0, 0, Yes], [3, 22, 23, 23, Yes], [3, 23, 22, 22, Yes], [4, 0, 20, 20, Yes], [4, 1, 19, 19, Yes], [4, 2, 18, 18, Yes], [4, 3, 17, 17, Yes], [4, 4, 16, 16, Yes], [4, 5, 15, 15, Yes], [4, 6, 14, 14, Yes], [4, 7, 13, 13, Yes], [4, 8, 12, 12, Yes], [4, 9, 11, 11, Yes], [4, 10, 10, 10, Yes], [4, 11, 9, 9, Yes], [4, 12, 8, 8, Yes], [4, 13, 7, 7, Yes], [4, 14, 6, 6, Yes], [4, 15, 5, 5, Yes], [4, 16, 4, 4, Yes], [4, 17, 3, 3, Yes], [4, 18, 2, 2, Yes], [4, 19, 1, 1, Yes], [4, 20, 0, 0, Yes], [4, 21, 23, 23, Yes], [4, 22, 22, 22, Yes], [4, 23, 21, 21, Yes], [5, 0, 19, 19, Yes], [5, 1, 18, 18, Yes], [5, 2, 17, 17, Yes], [5, 3, 16, 16, Yes], [5, 4, 15, 15, Yes], [5, 5, 14, 14, Yes], [5, 6, 13, 13, Yes], [5, 7, 12, 12, Yes], [5, 8, 11, 11, Yes], [5, 9, 10, 10, Yes], [5, 10, 9, 9, Yes], [5, 11, 8, 8, Yes], [5, 12, 7, 7, Yes], [5, 13, 6, 6, Yes], [5, 14, 5, 5, Yes], [5, 15, 4, 4, Yes], [5, 16, 3, 3, Yes], [5, 17, 2, 2, Yes], [5, 18, 1, 1, Yes], [5, 19, 0, 0, Yes], [5, 20, 23, 23, Yes], [5, 21, 22, 22, Yes], [5, 22, 21, 21, Yes], [5, 23, 20, 20, Yes], [6, 0, 18, 18, Yes], [6, 1, 17, 17, Yes], [6, 2, 16, 16, Yes], [6, 3, 15, 15, Yes], [6, 4, 14, 14, Yes], [6, 5, 13, 13, Yes], [6, 6, 12, 12, Yes], [6, 7, 11, 11, Yes], [6, 8, 10, 10, Yes], [6, 9, 9, 9, Yes], [6, 10, 8, 8, Yes], [6, 11, 7, 7, Yes], [6, 12, 6, 6, Yes], [6, 13, 5, 5, Yes], [6, 14, 4, 4, Yes], [6, 15, 3, 3, Yes], [6, 16, 2, 2, Yes], [6, 17, 1, 1, Yes], [6, 18, 0, 0, Yes], [6, 19, 23, 23, Yes], [6, 20, 22, 22, Yes], [6, 21, 21, 21, Yes], [6, 22, 20, 20, Yes], [6, 23, 19, 19, Yes], [7, 0, 17, 17, Yes], [7, 1, 16, 16, Yes], [7, 2, 15, 15, Yes], [7, 3, 14, 14, Yes], [7, 4, 13, 13, Yes], [7, 5, 12, 12, Yes], [7, 6, 11, 11, Yes], [7, 7, 10, 10, Yes], [7, 8, 9, 9, Yes], [7, 9, 8, 8, Yes], [7, 10, 7, 7, Yes], [7, 11, 6, 6, Yes], [7, 12, 5, 5, Yes], [7, 13, 4, 4, Yes], [7, 14, 3, 3, Yes], [7, 15, 2, 2, Yes], [7, 16, 1, 1, Yes], [7, 17, 0, 0, Yes], [7, 18, 23, 23, Yes], [7, 19, 22, 22, Yes], [7, 20, 21, 21, Yes], [7, 21, 20, 20, Yes], [7, 22, 19, 19, Yes], [7, 23, 18, 18, Yes], [8, 0, 16, 16, Yes], [8, 1, 15, 15, Yes], [8, 2, 14, 14, Yes], [8, 3, 13, 13, Yes], [8, 4, 12, 12, Yes], [8, 5, 11, 11, Yes], [8, 6, 10, 10, Yes], [8, 7, 9, 9, Yes], [8, 8, 8, 8, Yes], [8, 9, 7, 7, Yes], [8, 10, 6, 6, Yes], [8, 11, 5, 5, Yes], [8, 12, 4, 4, Yes], [8, 13, 3, 3, Yes], [8, 14, 2, 2, Yes], [8, 15, 1, 1, Yes], [8, 16, 0, 0, Yes], [8, 17, 23, 23, Yes], [8, 18, 22, 22, Yes], [8, 19, 21, 21, Yes], [8, 20, 20, 20, Yes], [8, 21, 19, 19, Yes], [8, 22, 18, 18, Yes], [8, 23, 17, 17, Yes], [9, 0, 15, 15, Yes], [9, 1, 14, 14, Yes], [9, 2, 13, 13, Yes], [9, 3, 12, 12, Yes], [9, 4, 11, 11, Yes], [9, 5, 10, 10, Yes], [9, 6, 9, 9, Yes], [9, 7, 8, 8, Yes], [9, 8, 7, 7, Yes], [9, 9, 6, 6, Yes], [9, 10, 5, 5, Yes], [9, 11, 4, 4, Yes], [9, 12, 3, 3, Yes], [9, 13, 2, 2, Yes], [9, 14, 1, 1, Yes], [9, 15, 0, 0, Yes], [9, 16, 23, 23, Yes], [9, 17, 22, 22, Yes], [9, 18, 21, 21, Yes], [9, 19, 20, 20, Yes], [9, 20, 19, 19, Yes], [9, 21, 18, 18, Yes], [9, 22, 17, 17, Yes], [9, 23, 16, 16, Yes], [10, 0, 14, 14, Yes], [10, 1, 13, 13, Yes], [10, 2, 12, 12, Yes], [10, 3, 11, 11, Yes], [10, 4, 10, 10, Yes], [10, 5, 9, 9, Yes], [10, 6, 8, 8, Yes], [10, 7, 7, 7, Yes], [10, 8, 6, 6, Yes], [10, 9, 5, 5, Yes], [10, 10, 4, 4, Yes], [10, 11, 3, 3, Yes], [10, 12, 2, 2, Yes], [10, 13, 1, 1, Yes], [10, 14, 0, 0, Yes], [10, 15, 23, 23, Yes], [10, 16, 22, 22, Yes], [10, 17, 21, 21, Yes], [10, 18, 20, 20, Yes], [10, 19, 19, 19, Yes], [10, 20, 18, 18, Yes], [10, 21, 17, 17, Yes], [10, 22, 16, 16, Yes], [10, 23, 15, 15, Yes], [11, 0, 13, 13, Yes], [11, 1, 12, 12, Yes], [11, 2, 11, 11, Yes], [11, 3, 10, 10, Yes], [11, 4, 9, 9, Yes], [11, 5, 8, 8, Yes], [11, 6, 7, 7, Yes], [11, 7, 6, 6, Yes], [11, 8, 5, 5, Yes], [11, 9, 4, 4, Yes], [11, 10, 3, 3, Yes], [11, 11, 2, 2, Yes], [11, 12, 1, 1, Yes], [11, 13, 0, 0, Yes], [11, 14, 23, 23, Yes], [11, 15, 22, 22, Yes], [11, 16, 21, 21, Yes], [11, 17, 20, 20, Yes], [11, 18, 19, 19, Yes], [11, 19, 18, 18, Yes], [11, 20, 17, 17, Yes], [11, 21, 16, 16, Yes], [11, 22, 15, 15, Yes], [11, 23, 14, 14, Yes], [12, 0, 12, 12, Yes], [12, 1, 11, 11, Yes], [12, 2, 10, 10, Yes], [12, 3, 9, 9, Yes], [12, 4, 8, 8, Yes], [12, 5, 7, 7, Yes], [12, 6, 6, 6, Yes], [12, 7, 5, 5, Yes], [12, 8, 4, 4, Yes], [12, 9, 3, 3, Yes], [12, 10, 2, 2, Yes], [12, 11, 1, 1, Yes], [12, 12, 0, 0, Yes], [12, 13, 23, 23, Yes], [12, 14, 22, 22, Yes], [12, 15, 21, 21, Yes], [12, 16, 20, 20, Yes], [12, 17, 19, 19, Yes], [12, 18, 18, 18, Yes], [12, 19, 17, 17, Yes], [12, 20, 16, 16, Yes], [12, 21, 15, 15, Yes], [12, 22, 14, 14, Yes], [12, 23, 13, 13, Yes], [13, 0, 11, 11, Yes], [13, 1, 10, 10, Yes], [13, 2, 9, 9, Yes], [13, 3, 8, 8, Yes], [13, 4, 7, 7, Yes], [13, 5, 6, 6, Yes], [13, 6, 5, 5, Yes], [13, 7, 4, 4, Yes], [13, 8, 3, 3, Yes], [13, 9, 2, 2, Yes], [13, 10, 1, 1, Yes], [13, 11, 0, 0, Yes], [13, 12, 23, 23, Yes], [13, 13, 22, 22, Yes], [13, 14, 21, 21, Yes], [13, 15, 20, 20, Yes], [13, 16, 19, 19, Yes], [13, 17, 18, 18, Yes], [13, 18, 17, 17, Yes], [13, 19, 16, 16, Yes], [13, 20, 15, 15, Yes], [13, 21, 14, 14, Yes], [13, 22, 13, 13, Yes], [13, 23, 12, 12, Yes], [14, 0, 10, 10, Yes], [14, 1, 9, 9, Yes], [14, 2, 8, 8, Yes], [14, 3, 7, 7, Yes], [14, 4, 6, 6, Yes], [14, 5, 5, 5, Yes], [14, 6, 4, 4, Yes], [14, 7, 3, 3, Yes], [14, 8, 2, 2, Yes], [14, 9, 1, 1, Yes], [14, 10, 0, 0, Yes], [14, 11, 23, 23, Yes], [14, 12, 22, 22, Yes], [14, 13, 21, 21, Yes], [14, 14, 20, 20, Yes], [14, 15, 19, 19, Yes], [14, 16, 18, 18, Yes], [14, 17, 17, 17, Yes], [14, 18, 16, 16, Yes], [14, 19, 15, 15, Yes], [14, 20, 14, 14, Yes], [14, 21, 13, 13, Yes], [14, 22, 12, 12, Yes], [14, 23, 11, 11, Yes], [15, 0, 9, 9, Yes], [15, 1, 8, 8, Yes], [15, 2, 7, 7, Yes], [15, 3, 6, 6, Yes], [15, 4, 5, 5, Yes], [15, 5, 4, 4, Yes], [15, 6, 3, 3, Yes], [15, 7, 2, 2, Yes], [15, 8, 1, 1, Yes], [15, 9, 0, 0, Yes], [15, 10, 23, 23, Yes], [15, 11, 22, 22, Yes], [15, 12, 21, 21, Yes], [15, 13, 20, 20, Yes], [15, 14, 19, 19, Yes], [15, 15, 18, 18, Yes], [15, 16, 17, 17, Yes], [15, 17, 16, 16, Yes], [15, 18, 15, 15, Yes], [15, 19, 14, 14, Yes], [15, 20, 13, 13, Yes], [15, 21, 12, 12, Yes], [15, 22, 11, 11, Yes], [15, 23, 10, 10, Yes], [16, 0, 8, 8, Yes], [16, 1, 7, 7, Yes], [16, 2, 6, 6, Yes], [16, 3, 5, 5, Yes], [16, 4, 4, 4, Yes], [16, 5, 3, 3, Yes], [16, 6, 2, 2, Yes], [16, 7, 1, 1, Yes], [16, 8, 0, 0, Yes], [16, 9, 23, 23, Yes], [16, 10, 22, 22, Yes], [16, 11, 21, 21, Yes], [16, 12, 20, 20, Yes], [16, 13, 19, 19, Yes], [16, 14, 18, 18, Yes], [16, 15, 17, 17, Yes], [16, 16, 16, 16, Yes], [16, 17, 15, 15, Yes], [16, 18, 14, 14, Yes], [16, 19, 13, 13, Yes], [16, 20, 12, 12, Yes], [16, 21, 11, 11, Yes], [16, 22, 10, 10, Yes], [16, 23, 9, 9, Yes], [17, 0, 7, 7, Yes], [17, 1, 6, 6, Yes], [17, 2, 5, 5, Yes], [17, 3, 4, 4, Yes], [17, 4, 3, 3, Yes], [17, 5, 2, 2, Yes], [17, 6, 1, 1, Yes], [17, 7, 0, 0, Yes], [17, 8, 23, 23, Yes], [17, 9, 22, 22, Yes], [17, 10, 21, 21, Yes], [17, 11, 20, 20, Yes], [17, 12, 19, 19, Yes], [17, 13, 18, 18, Yes], [17, 14, 17, 17, Yes], [17, 15, 16, 16, Yes], [17, 16, 15, 15, Yes], [17, 17, 14, 14, Yes], [17, 18, 13, 13, Yes], [17, 19, 12, 12, Yes], [17, 20, 11, 11, Yes], [17, 21, 10, 10, Yes], [17, 22, 9, 9, Yes], [17, 23, 8, 8, Yes], [18, 0, 6, 6, Yes], [18, 1, 5, 5, Yes], [18, 2, 4, 4, Yes], [18, 3, 3, 3, Yes], [18, 4, 2, 2, Yes], [18, 5, 1, 1, Yes], [18, 6, 0, 0, Yes], [18, 7, 23, 23, Yes], [18, 8, 22, 22, Yes], [18, 9, 21, 21, Yes], [18, 10, 20, 20, Yes], [18, 11, 19, 19, Yes], [18, 12, 18, 18, Yes], [18, 13, 17, 17, Yes], [18, 14, 16, 16, Yes], [18, 15, 15, 15, Yes], [18, 16, 14, 14, Yes], [18, 17, 13, 13, Yes], [18, 18, 12, 12, Yes], [18, 19, 11, 11, Yes], [18, 20, 10, 10, Yes], [18, 21, 9, 9, Yes], [18, 22, 8, 8, Yes], [18, 23, 7, 7, Yes], [19, 0, 5, 5, Yes], [19, 1, 4, 4, Yes], [19, 2, 3, 3, Yes], [19, 3, 2, 2, Yes], [19, 4, 1, 1, Yes], [19, 5, 0, 0, Yes], [19, 6, 23, 23, Yes], [19, 7, 22, 22, Yes], [19, 8, 21, 21, Yes], [19, 9, 20, 20, Yes], [19, 10, 19, 19, Yes], [19, 11, 18, 18, Yes], [19, 12, 17, 17, Yes], [19, 13, 16, 16, Yes], [19, 14, 15, 15, Yes], [19, 15, 14, 14, Yes], [19, 16, 13, 13, Yes], [19, 17, 12, 12, Yes], [19, 18, 11, 11, Yes], [19, 19, 10, 10, Yes], [19, 20, 9, 9, Yes], [19, 21, 8, 8, Yes], [19, 22, 7, 7, Yes], [19, 23, 6, 6, Yes], [20, 0, 4, 4, Yes], [20, 1, 3, 3, Yes], [20, 2, 2, 2, Yes], [20, 3, 1, 1, Yes], [20, 4, 0, 0, Yes], [20, 5, 23, 23, Yes], [20, 6, 22, 22, Yes], [20, 7, 21, 21, Yes], [20, 8, 20, 20, Yes], [20, 9, 19, 19, Yes], [20, 10, 18, 18, Yes], [20, 11, 17, 17, Yes], [20, 12, 16, 16, Yes], [20, 13, 15, 15, Yes], [20, 14, 14, 14, Yes], [20, 15, 13, 13, Yes], [20, 16, 12, 12, Yes], [20, 17, 11, 11, Yes], [20, 18, 10, 10, Yes], [20, 19, 9, 9, Yes], [20, 20, 8, 8, Yes], [20, 21, 7, 7, Yes], [20, 22, 6, 6, Yes], [20, 23, 5, 5, Yes], [21, 0, 3, 3, Yes], [21, 1, 2, 2, Yes], [21, 2, 1, 1, Yes], [21, 3, 0, 0, Yes], [21, 4, 23, 23, Yes], [21, 5, 22, 22, Yes], [21, 6, 21, 21, Yes], [21, 7, 20, 20, Yes], [21, 8, 19, 19, Yes], [21, 9, 18, 18, Yes], [21, 10, 17, 17, Yes], [21, 11, 16, 16, Yes], [21, 12, 15, 15, Yes], [21, 13, 14, 14, Yes], [21, 14, 13, 13, Yes], [21, 15, 12, 12, Yes], [21, 16, 11, 11, Yes], [21, 17, 10, 10, Yes], [21, 18, 9, 9, Yes], [21, 19, 8, 8, Yes], [21, 20, 7, 7, Yes], [21, 21, 6, 6, Yes], [21, 22, 5, 5, Yes], [21, 23, 4, 4, Yes], [22, 0, 2, 2, Yes], [22, 1, 1, 1, Yes], [22, 2, 0, 0, Yes], [22, 3, 23, 23, Yes], [22, 4, 22, 22, Yes], [22, 5, 21, 21, Yes], [22, 6, 20, 20, Yes], [22, 7, 19, 19, Yes], [22, 8, 18, 18, Yes], [22, 9, 17, 17, Yes], [22, 10, 16, 16, Yes], [22, 11, 15, 15, Yes], [22, 12, 14, 14, Yes], [22, 13, 13, 13, Yes], [22, 14, 12, 12, Yes], [22, 15, 11, 11, Yes], [22, 16, 10, 10, Yes], [22, 17, 9, 9, Yes], [22, 18, 8, 8, Yes], [22, 19, 7, 7, Yes], [22, 20, 6, 6, Yes], [22, 21, 5, 5, Yes], [22, 22, 4, 4, Yes], [22, 23, 3, 3, Yes], [23, 0, 1, 1, Yes], [23, 1, 0, 0, Yes], [23, 2, 23, 23, Yes], [23, 3, 22, 22, Yes], [23, 4, 21, 21, Yes], [23, 5, 20, 20, Yes], [23, 6, 19, 19, Yes], [23, 7, 18, 18, Yes], [23, 8, 17, 17, Yes], [23, 9, 16, 16, Yes], [23, 10, 15, 15, Yes], [23, 11, 14, 14, Yes], [23, 12, 13, 13, Yes], [23, 13, 12, 12, Yes], [23, 14, 11, 11, Yes], [23, 15, 10, 10, Yes], [23, 16, 9, 9, Yes], [23, 17, 8, 8, Yes], [23, 18, 7, 7, Yes], [23, 19, 6, 6, Yes], [23, 20, 5, 5, Yes], [23, 21, 4, 4, Yes], [23, 22, 3, 3, Yes], [23, 23, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-homomorphism-qualification · candidate
23. Qualify Both Groups and Every Product Pair before Applying Kernel Theorems
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]], target identity=0, and total mapping=[0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).
View answer and explanation
(25, 25, [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 24, 24, Yes], [0, 2, 23, 23, Yes], [0, 3, 22, 22, Yes], [0, 4, 21, 21, Yes], [0, 5, 20, 20, Yes], [0, 6, 19, 19, Yes], [0, 7, 18, 18, Yes], [0, 8, 17, 17, Yes], [0, 9, 16, 16, Yes], [0, 10, 15, 15, Yes], [0, 11, 14, 14, Yes], [0, 12, 13, 13, Yes], [0, 13, 12, 12, Yes], [0, 14, 11, 11, Yes], [0, 15, 10, 10, Yes], [0, 16, 9, 9, Yes], [0, 17, 8, 8, Yes], [0, 18, 7, 7, Yes], [0, 19, 6, 6, Yes], [0, 20, 5, 5, Yes], [0, 21, 4, 4, Yes], [0, 22, 3, 3, Yes], [0, 23, 2, 2, Yes], [0, 24, 1, 1, Yes], [1, 0, 24, 24, Yes], [1, 1, 23, 23, Yes], [1, 2, 22, 22, Yes], [1, 3, 21, 21, Yes], [1, 4, 20, 20, Yes], [1, 5, 19, 19, Yes], [1, 6, 18, 18, Yes], [1, 7, 17, 17, Yes], [1, 8, 16, 16, Yes], [1, 9, 15, 15, Yes], [1, 10, 14, 14, Yes], [1, 11, 13, 13, Yes], [1, 12, 12, 12, Yes], [1, 13, 11, 11, Yes], [1, 14, 10, 10, Yes], [1, 15, 9, 9, Yes], [1, 16, 8, 8, Yes], [1, 17, 7, 7, Yes], [1, 18, 6, 6, Yes], [1, 19, 5, 5, Yes], [1, 20, 4, 4, Yes], [1, 21, 3, 3, Yes], [1, 22, 2, 2, Yes], [1, 23, 1, 1, Yes], [1, 24, 0, 0, Yes], [2, 0, 23, 23, Yes], [2, 1, 22, 22, Yes], [2, 2, 21, 21, Yes], [2, 3, 20, 20, Yes], [2, 4, 19, 19, Yes], [2, 5, 18, 18, Yes], [2, 6, 17, 17, Yes], [2, 7, 16, 16, Yes], [2, 8, 15, 15, Yes], [2, 9, 14, 14, Yes], [2, 10, 13, 13, Yes], [2, 11, 12, 12, Yes], [2, 12, 11, 11, Yes], [2, 13, 10, 10, Yes], [2, 14, 9, 9, Yes], [2, 15, 8, 8, Yes], [2, 16, 7, 7, Yes], [2, 17, 6, 6, Yes], [2, 18, 5, 5, Yes], [2, 19, 4, 4, Yes], [2, 20, 3, 3, Yes], [2, 21, 2, 2, Yes], [2, 22, 1, 1, Yes], [2, 23, 0, 0, Yes], [2, 24, 24, 24, Yes], [3, 0, 22, 22, Yes], [3, 1, 21, 21, Yes], [3, 2, 20, 20, Yes], [3, 3, 19, 19, Yes], [3, 4, 18, 18, Yes], [3, 5, 17, 17, Yes], [3, 6, 16, 16, Yes], [3, 7, 15, 15, Yes], [3, 8, 14, 14, Yes], [3, 9, 13, 13, Yes], [3, 10, 12, 12, Yes], [3, 11, 11, 11, Yes], [3, 12, 10, 10, Yes], [3, 13, 9, 9, Yes], [3, 14, 8, 8, Yes], [3, 15, 7, 7, Yes], [3, 16, 6, 6, Yes], [3, 17, 5, 5, Yes], [3, 18, 4, 4, Yes], [3, 19, 3, 3, Yes], [3, 20, 2, 2, Yes], [3, 21, 1, 1, Yes], [3, 22, 0, 0, Yes], [3, 23, 24, 24, Yes], [3, 24, 23, 23, Yes], [4, 0, 21, 21, Yes], [4, 1, 20, 20, Yes], [4, 2, 19, 19, Yes], [4, 3, 18, 18, Yes], [4, 4, 17, 17, Yes], [4, 5, 16, 16, Yes], [4, 6, 15, 15, Yes], [4, 7, 14, 14, Yes], [4, 8, 13, 13, Yes], [4, 9, 12, 12, Yes], [4, 10, 11, 11, Yes], [4, 11, 10, 10, Yes], [4, 12, 9, 9, Yes], [4, 13, 8, 8, Yes], [4, 14, 7, 7, Yes], [4, 15, 6, 6, Yes], [4, 16, 5, 5, Yes], [4, 17, 4, 4, Yes], [4, 18, 3, 3, Yes], [4, 19, 2, 2, Yes], [4, 20, 1, 1, Yes], [4, 21, 0, 0, Yes], [4, 22, 24, 24, Yes], [4, 23, 23, 23, Yes], [4, 24, 22, 22, Yes], [5, 0, 20, 20, Yes], [5, 1, 19, 19, Yes], [5, 2, 18, 18, Yes], [5, 3, 17, 17, Yes], [5, 4, 16, 16, Yes], [5, 5, 15, 15, Yes], [5, 6, 14, 14, Yes], [5, 7, 13, 13, Yes], [5, 8, 12, 12, Yes], [5, 9, 11, 11, Yes], [5, 10, 10, 10, Yes], [5, 11, 9, 9, Yes], [5, 12, 8, 8, Yes], [5, 13, 7, 7, Yes], [5, 14, 6, 6, Yes], [5, 15, 5, 5, Yes], [5, 16, 4, 4, Yes], [5, 17, 3, 3, Yes], [5, 18, 2, 2, Yes], [5, 19, 1, 1, Yes], [5, 20, 0, 0, Yes], [5, 21, 24, 24, Yes], [5, 22, 23, 23, Yes], [5, 23, 22, 22, Yes], [5, 24, 21, 21, Yes], [6, 0, 19, 19, Yes], [6, 1, 18, 18, Yes], [6, 2, 17, 17, Yes], [6, 3, 16, 16, Yes], [6, 4, 15, 15, Yes], [6, 5, 14, 14, Yes], [6, 6, 13, 13, Yes], [6, 7, 12, 12, Yes], [6, 8, 11, 11, Yes], [6, 9, 10, 10, Yes], [6, 10, 9, 9, Yes], [6, 11, 8, 8, Yes], [6, 12, 7, 7, Yes], [6, 13, 6, 6, Yes], [6, 14, 5, 5, Yes], [6, 15, 4, 4, Yes], [6, 16, 3, 3, Yes], [6, 17, 2, 2, Yes], [6, 18, 1, 1, Yes], [6, 19, 0, 0, Yes], [6, 20, 24, 24, Yes], [6, 21, 23, 23, Yes], [6, 22, 22, 22, Yes], [6, 23, 21, 21, Yes], [6, 24, 20, 20, Yes], [7, 0, 18, 18, Yes], [7, 1, 17, 17, Yes], [7, 2, 16, 16, Yes], [7, 3, 15, 15, Yes], [7, 4, 14, 14, Yes], [7, 5, 13, 13, Yes], [7, 6, 12, 12, Yes], [7, 7, 11, 11, Yes], [7, 8, 10, 10, Yes], [7, 9, 9, 9, Yes], [7, 10, 8, 8, Yes], [7, 11, 7, 7, Yes], [7, 12, 6, 6, Yes], [7, 13, 5, 5, Yes], [7, 14, 4, 4, Yes], [7, 15, 3, 3, Yes], [7, 16, 2, 2, Yes], [7, 17, 1, 1, Yes], [7, 18, 0, 0, Yes], [7, 19, 24, 24, Yes], [7, 20, 23, 23, Yes], [7, 21, 22, 22, Yes], [7, 22, 21, 21, Yes], [7, 23, 20, 20, Yes], [7, 24, 19, 19, Yes], [8, 0, 17, 17, Yes], [8, 1, 16, 16, Yes], [8, 2, 15, 15, Yes], [8, 3, 14, 14, Yes], [8, 4, 13, 13, Yes], [8, 5, 12, 12, Yes], [8, 6, 11, 11, Yes], [8, 7, 10, 10, Yes], [8, 8, 9, 9, Yes], [8, 9, 8, 8, Yes], [8, 10, 7, 7, Yes], [8, 11, 6, 6, Yes], [8, 12, 5, 5, Yes], [8, 13, 4, 4, Yes], [8, 14, 3, 3, Yes], [8, 15, 2, 2, Yes], [8, 16, 1, 1, Yes], [8, 17, 0, 0, Yes], [8, 18, 24, 24, Yes], [8, 19, 23, 23, Yes], [8, 20, 22, 22, Yes], [8, 21, 21, 21, Yes], [8, 22, 20, 20, Yes], [8, 23, 19, 19, Yes], [8, 24, 18, 18, Yes], [9, 0, 16, 16, Yes], [9, 1, 15, 15, Yes], [9, 2, 14, 14, Yes], [9, 3, 13, 13, Yes], [9, 4, 12, 12, Yes], [9, 5, 11, 11, Yes], [9, 6, 10, 10, Yes], [9, 7, 9, 9, Yes], [9, 8, 8, 8, Yes], [9, 9, 7, 7, Yes], [9, 10, 6, 6, Yes], [9, 11, 5, 5, Yes], [9, 12, 4, 4, Yes], [9, 13, 3, 3, Yes], [9, 14, 2, 2, Yes], [9, 15, 1, 1, Yes], [9, 16, 0, 0, Yes], [9, 17, 24, 24, Yes], [9, 18, 23, 23, Yes], [9, 19, 22, 22, Yes], [9, 20, 21, 21, Yes], [9, 21, 20, 20, Yes], [9, 22, 19, 19, Yes], [9, 23, 18, 18, Yes], [9, 24, 17, 17, Yes], [10, 0, 15, 15, Yes], [10, 1, 14, 14, Yes], [10, 2, 13, 13, Yes], [10, 3, 12, 12, Yes], [10, 4, 11, 11, Yes], [10, 5, 10, 10, Yes], [10, 6, 9, 9, Yes], [10, 7, 8, 8, Yes], [10, 8, 7, 7, Yes], [10, 9, 6, 6, Yes], [10, 10, 5, 5, Yes], [10, 11, 4, 4, Yes], [10, 12, 3, 3, Yes], [10, 13, 2, 2, Yes], [10, 14, 1, 1, Yes], [10, 15, 0, 0, Yes], [10, 16, 24, 24, Yes], [10, 17, 23, 23, Yes], [10, 18, 22, 22, Yes], [10, 19, 21, 21, Yes], [10, 20, 20, 20, Yes], [10, 21, 19, 19, Yes], [10, 22, 18, 18, Yes], [10, 23, 17, 17, Yes], [10, 24, 16, 16, Yes], [11, 0, 14, 14, Yes], [11, 1, 13, 13, Yes], [11, 2, 12, 12, Yes], [11, 3, 11, 11, Yes], [11, 4, 10, 10, Yes], [11, 5, 9, 9, Yes], [11, 6, 8, 8, Yes], [11, 7, 7, 7, Yes], [11, 8, 6, 6, Yes], [11, 9, 5, 5, Yes], [11, 10, 4, 4, Yes], [11, 11, 3, 3, Yes], [11, 12, 2, 2, Yes], [11, 13, 1, 1, Yes], [11, 14, 0, 0, Yes], [11, 15, 24, 24, Yes], [11, 16, 23, 23, Yes], [11, 17, 22, 22, Yes], [11, 18, 21, 21, Yes], [11, 19, 20, 20, Yes], [11, 20, 19, 19, Yes], [11, 21, 18, 18, Yes], [11, 22, 17, 17, Yes], [11, 23, 16, 16, Yes], [11, 24, 15, 15, Yes], [12, 0, 13, 13, Yes], [12, 1, 12, 12, Yes], [12, 2, 11, 11, Yes], [12, 3, 10, 10, Yes], [12, 4, 9, 9, Yes], [12, 5, 8, 8, Yes], [12, 6, 7, 7, Yes], [12, 7, 6, 6, Yes], [12, 8, 5, 5, Yes], [12, 9, 4, 4, Yes], [12, 10, 3, 3, Yes], [12, 11, 2, 2, Yes], [12, 12, 1, 1, Yes], [12, 13, 0, 0, Yes], [12, 14, 24, 24, Yes], [12, 15, 23, 23, Yes], [12, 16, 22, 22, Yes], [12, 17, 21, 21, Yes], [12, 18, 20, 20, Yes], [12, 19, 19, 19, Yes], [12, 20, 18, 18, Yes], [12, 21, 17, 17, Yes], [12, 22, 16, 16, Yes], [12, 23, 15, 15, Yes], [12, 24, 14, 14, Yes], [13, 0, 12, 12, Yes], [13, 1, 11, 11, Yes], [13, 2, 10, 10, Yes], [13, 3, 9, 9, Yes], [13, 4, 8, 8, Yes], [13, 5, 7, 7, Yes], [13, 6, 6, 6, Yes], [13, 7, 5, 5, Yes], [13, 8, 4, 4, Yes], [13, 9, 3, 3, Yes], [13, 10, 2, 2, Yes], [13, 11, 1, 1, Yes], [13, 12, 0, 0, Yes], [13, 13, 24, 24, Yes], [13, 14, 23, 23, Yes], [13, 15, 22, 22, Yes], [13, 16, 21, 21, Yes], [13, 17, 20, 20, Yes], [13, 18, 19, 19, Yes], [13, 19, 18, 18, Yes], [13, 20, 17, 17, Yes], [13, 21, 16, 16, Yes], [13, 22, 15, 15, Yes], [13, 23, 14, 14, Yes], [13, 24, 13, 13, Yes], [14, 0, 11, 11, Yes], [14, 1, 10, 10, Yes], [14, 2, 9, 9, Yes], [14, 3, 8, 8, Yes], [14, 4, 7, 7, Yes], [14, 5, 6, 6, Yes], [14, 6, 5, 5, Yes], [14, 7, 4, 4, Yes], [14, 8, 3, 3, Yes], [14, 9, 2, 2, Yes], [14, 10, 1, 1, Yes], [14, 11, 0, 0, Yes], [14, 12, 24, 24, Yes], [14, 13, 23, 23, Yes], [14, 14, 22, 22, Yes], [14, 15, 21, 21, Yes], [14, 16, 20, 20, Yes], [14, 17, 19, 19, Yes], [14, 18, 18, 18, Yes], [14, 19, 17, 17, Yes], [14, 20, 16, 16, Yes], [14, 21, 15, 15, Yes], [14, 22, 14, 14, Yes], [14, 23, 13, 13, Yes], [14, 24, 12, 12, Yes], [15, 0, 10, 10, Yes], [15, 1, 9, 9, Yes], [15, 2, 8, 8, Yes], [15, 3, 7, 7, Yes], [15, 4, 6, 6, Yes], [15, 5, 5, 5, Yes], [15, 6, 4, 4, Yes], [15, 7, 3, 3, Yes], [15, 8, 2, 2, Yes], [15, 9, 1, 1, Yes], [15, 10, 0, 0, Yes], [15, 11, 24, 24, Yes], [15, 12, 23, 23, Yes], [15, 13, 22, 22, Yes], [15, 14, 21, 21, Yes], [15, 15, 20, 20, Yes], [15, 16, 19, 19, Yes], [15, 17, 18, 18, Yes], [15, 18, 17, 17, Yes], [15, 19, 16, 16, Yes], [15, 20, 15, 15, Yes], [15, 21, 14, 14, Yes], [15, 22, 13, 13, Yes], [15, 23, 12, 12, Yes], [15, 24, 11, 11, Yes], [16, 0, 9, 9, Yes], [16, 1, 8, 8, Yes], [16, 2, 7, 7, Yes], [16, 3, 6, 6, Yes], [16, 4, 5, 5, Yes], [16, 5, 4, 4, Yes], [16, 6, 3, 3, Yes], [16, 7, 2, 2, Yes], [16, 8, 1, 1, Yes], [16, 9, 0, 0, Yes], [16, 10, 24, 24, Yes], [16, 11, 23, 23, Yes], [16, 12, 22, 22, Yes], [16, 13, 21, 21, Yes], [16, 14, 20, 20, Yes], [16, 15, 19, 19, Yes], [16, 16, 18, 18, Yes], [16, 17, 17, 17, Yes], [16, 18, 16, 16, Yes], [16, 19, 15, 15, Yes], [16, 20, 14, 14, Yes], [16, 21, 13, 13, Yes], [16, 22, 12, 12, Yes], [16, 23, 11, 11, Yes], [16, 24, 10, 10, Yes], [17, 0, 8, 8, Yes], [17, 1, 7, 7, Yes], [17, 2, 6, 6, Yes], [17, 3, 5, 5, Yes], [17, 4, 4, 4, Yes], [17, 5, 3, 3, Yes], [17, 6, 2, 2, Yes], [17, 7, 1, 1, Yes], [17, 8, 0, 0, Yes], [17, 9, 24, 24, Yes], [17, 10, 23, 23, Yes], [17, 11, 22, 22, Yes], [17, 12, 21, 21, Yes], [17, 13, 20, 20, Yes], [17, 14, 19, 19, Yes], [17, 15, 18, 18, Yes], [17, 16, 17, 17, Yes], [17, 17, 16, 16, Yes], [17, 18, 15, 15, Yes], [17, 19, 14, 14, Yes], [17, 20, 13, 13, Yes], [17, 21, 12, 12, Yes], [17, 22, 11, 11, Yes], [17, 23, 10, 10, Yes], [17, 24, 9, 9, Yes], [18, 0, 7, 7, Yes], [18, 1, 6, 6, Yes], [18, 2, 5, 5, Yes], [18, 3, 4, 4, Yes], [18, 4, 3, 3, Yes], [18, 5, 2, 2, Yes], [18, 6, 1, 1, Yes], [18, 7, 0, 0, Yes], [18, 8, 24, 24, Yes], [18, 9, 23, 23, Yes], [18, 10, 22, 22, Yes], [18, 11, 21, 21, Yes], [18, 12, 20, 20, Yes], [18, 13, 19, 19, Yes], [18, 14, 18, 18, Yes], [18, 15, 17, 17, Yes], [18, 16, 16, 16, Yes], [18, 17, 15, 15, Yes], [18, 18, 14, 14, Yes], [18, 19, 13, 13, Yes], [18, 20, 12, 12, Yes], [18, 21, 11, 11, Yes], [18, 22, 10, 10, Yes], [18, 23, 9, 9, Yes], [18, 24, 8, 8, Yes], [19, 0, 6, 6, Yes], [19, 1, 5, 5, Yes], [19, 2, 4, 4, Yes], [19, 3, 3, 3, Yes], [19, 4, 2, 2, Yes], [19, 5, 1, 1, Yes], [19, 6, 0, 0, Yes], [19, 7, 24, 24, Yes], [19, 8, 23, 23, Yes], [19, 9, 22, 22, Yes], [19, 10, 21, 21, Yes], [19, 11, 20, 20, Yes], [19, 12, 19, 19, Yes], [19, 13, 18, 18, Yes], [19, 14, 17, 17, Yes], [19, 15, 16, 16, Yes], [19, 16, 15, 15, Yes], [19, 17, 14, 14, Yes], [19, 18, 13, 13, Yes], [19, 19, 12, 12, Yes], [19, 20, 11, 11, Yes], [19, 21, 10, 10, Yes], [19, 22, 9, 9, Yes], [19, 23, 8, 8, Yes], [19, 24, 7, 7, Yes], [20, 0, 5, 5, Yes], [20, 1, 4, 4, Yes], [20, 2, 3, 3, Yes], [20, 3, 2, 2, Yes], [20, 4, 1, 1, Yes], [20, 5, 0, 0, Yes], [20, 6, 24, 24, Yes], [20, 7, 23, 23, Yes], [20, 8, 22, 22, Yes], [20, 9, 21, 21, Yes], [20, 10, 20, 20, Yes], [20, 11, 19, 19, Yes], [20, 12, 18, 18, Yes], [20, 13, 17, 17, Yes], [20, 14, 16, 16, Yes], [20, 15, 15, 15, Yes], [20, 16, 14, 14, Yes], [20, 17, 13, 13, Yes], [20, 18, 12, 12, Yes], [20, 19, 11, 11, Yes], [20, 20, 10, 10, Yes], [20, 21, 9, 9, Yes], [20, 22, 8, 8, Yes], [20, 23, 7, 7, Yes], [20, 24, 6, 6, Yes], [21, 0, 4, 4, Yes], [21, 1, 3, 3, Yes], [21, 2, 2, 2, Yes], [21, 3, 1, 1, Yes], [21, 4, 0, 0, Yes], [21, 5, 24, 24, Yes], [21, 6, 23, 23, Yes], [21, 7, 22, 22, Yes], [21, 8, 21, 21, Yes], [21, 9, 20, 20, Yes], [21, 10, 19, 19, Yes], [21, 11, 18, 18, Yes], [21, 12, 17, 17, Yes], [21, 13, 16, 16, Yes], [21, 14, 15, 15, Yes], [21, 15, 14, 14, Yes], [21, 16, 13, 13, Yes], [21, 17, 12, 12, Yes], [21, 18, 11, 11, Yes], [21, 19, 10, 10, Yes], [21, 20, 9, 9, Yes], [21, 21, 8, 8, Yes], [21, 22, 7, 7, Yes], [21, 23, 6, 6, Yes], [21, 24, 5, 5, Yes], [22, 0, 3, 3, Yes], [22, 1, 2, 2, Yes], [22, 2, 1, 1, Yes], [22, 3, 0, 0, Yes], [22, 4, 24, 24, Yes], [22, 5, 23, 23, Yes], [22, 6, 22, 22, Yes], [22, 7, 21, 21, Yes], [22, 8, 20, 20, Yes], [22, 9, 19, 19, Yes], [22, 10, 18, 18, Yes], [22, 11, 17, 17, Yes], [22, 12, 16, 16, Yes], [22, 13, 15, 15, Yes], [22, 14, 14, 14, Yes], [22, 15, 13, 13, Yes], [22, 16, 12, 12, Yes], [22, 17, 11, 11, Yes], [22, 18, 10, 10, Yes], [22, 19, 9, 9, Yes], [22, 20, 8, 8, Yes], [22, 21, 7, 7, Yes], [22, 22, 6, 6, Yes], [22, 23, 5, 5, Yes], [22, 24, 4, 4, Yes], [23, 0, 2, 2, Yes], [23, 1, 1, 1, Yes], [23, 2, 0, 0, Yes], [23, 3, 24, 24, Yes], [23, 4, 23, 23, Yes], [23, 5, 22, 22, Yes], [23, 6, 21, 21, Yes], [23, 7, 20, 20, Yes], [23, 8, 19, 19, Yes], [23, 9, 18, 18, Yes], [23, 10, 17, 17, Yes], [23, 11, 16, 16, Yes], [23, 12, 15, 15, Yes], [23, 13, 14, 14, Yes], [23, 14, 13, 13, Yes], [23, 15, 12, 12, Yes], [23, 16, 11, 11, Yes], [23, 17, 10, 10, Yes], [23, 18, 9, 9, Yes], [23, 19, 8, 8, Yes], [23, 20, 7, 7, Yes], [23, 21, 6, 6, Yes], [23, 22, 5, 5, Yes], [23, 23, 4, 4, Yes], [23, 24, 3, 3, Yes], [24, 0, 1, 1, Yes], [24, 1, 0, 0, Yes], [24, 2, 24, 24, Yes], [24, 3, 23, 23, Yes], [24, 4, 22, 22, Yes], [24, 5, 21, 21, Yes], [24, 6, 20, 20, Yes], [24, 7, 19, 19, Yes], [24, 8, 18, 18, Yes], [24, 9, 17, 17, Yes], [24, 10, 16, 16, Yes], [24, 11, 15, 15, Yes], [24, 12, 14, 14, Yes], [24, 13, 13, 13, Yes], [24, 14, 12, 12, Yes], [24, 15, 11, 11, Yes], [24, 16, 10, 10, Yes], [24, 17, 9, 9, Yes], [24, 18, 8, 8, Yes], [24, 19, 7, 7, Yes], [24, 20, 6, 6, Yes], [24, 21, 5, 5, Yes], [24, 22, 4, 4, Yes], [24, 23, 3, 3, Yes], [24, 24, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)
List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (25, 25, [0, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 24, 24, Yes], [0, 2, 23, 23, Yes], [0, 3, 22, 22, Yes], [0, 4, 21, 21, Yes], [0, 5, 20, 20, Yes], [0, 6, 19, 19, Yes], [0, 7, 18, 18, Yes], [0, 8, 17, 17, Yes], [0, 9, 16, 16, Yes], [0, 10, 15, 15, Yes], [0, 11, 14, 14, Yes], [0, 12, 13, 13, Yes], [0, 13, 12, 12, Yes], [0, 14, 11, 11, Yes], [0, 15, 10, 10, Yes], [0, 16, 9, 9, Yes], [0, 17, 8, 8, Yes], [0, 18, 7, 7, Yes], [0, 19, 6, 6, Yes], [0, 20, 5, 5, Yes], [0, 21, 4, 4, Yes], [0, 22, 3, 3, Yes], [0, 23, 2, 2, Yes], [0, 24, 1, 1, Yes], [1, 0, 24, 24, Yes], [1, 1, 23, 23, Yes], [1, 2, 22, 22, Yes], [1, 3, 21, 21, Yes], [1, 4, 20, 20, Yes], [1, 5, 19, 19, Yes], [1, 6, 18, 18, Yes], [1, 7, 17, 17, Yes], [1, 8, 16, 16, Yes], [1, 9, 15, 15, Yes], [1, 10, 14, 14, Yes], [1, 11, 13, 13, Yes], [1, 12, 12, 12, Yes], [1, 13, 11, 11, Yes], [1, 14, 10, 10, Yes], [1, 15, 9, 9, Yes], [1, 16, 8, 8, Yes], [1, 17, 7, 7, Yes], [1, 18, 6, 6, Yes], [1, 19, 5, 5, Yes], [1, 20, 4, 4, Yes], [1, 21, 3, 3, Yes], [1, 22, 2, 2, Yes], [1, 23, 1, 1, Yes], [1, 24, 0, 0, Yes], [2, 0, 23, 23, Yes], [2, 1, 22, 22, Yes], [2, 2, 21, 21, Yes], [2, 3, 20, 20, Yes], [2, 4, 19, 19, Yes], [2, 5, 18, 18, Yes], [2, 6, 17, 17, Yes], [2, 7, 16, 16, Yes], [2, 8, 15, 15, Yes], [2, 9, 14, 14, Yes], [2, 10, 13, 13, Yes], [2, 11, 12, 12, Yes], [2, 12, 11, 11, Yes], [2, 13, 10, 10, Yes], [2, 14, 9, 9, Yes], [2, 15, 8, 8, Yes], [2, 16, 7, 7, Yes], [2, 17, 6, 6, Yes], [2, 18, 5, 5, Yes], [2, 19, 4, 4, Yes], [2, 20, 3, 3, Yes], [2, 21, 2, 2, Yes], [2, 22, 1, 1, Yes], [2, 23, 0, 0, Yes], [2, 24, 24, 24, Yes], [3, 0, 22, 22, Yes], [3, 1, 21, 21, Yes], [3, 2, 20, 20, Yes], [3, 3, 19, 19, Yes], [3, 4, 18, 18, Yes], [3, 5, 17, 17, Yes], [3, 6, 16, 16, Yes], [3, 7, 15, 15, Yes], [3, 8, 14, 14, Yes], [3, 9, 13, 13, Yes], [3, 10, 12, 12, Yes], [3, 11, 11, 11, Yes], [3, 12, 10, 10, Yes], [3, 13, 9, 9, Yes], [3, 14, 8, 8, Yes], [3, 15, 7, 7, Yes], [3, 16, 6, 6, Yes], [3, 17, 5, 5, Yes], [3, 18, 4, 4, Yes], [3, 19, 3, 3, Yes], [3, 20, 2, 2, Yes], [3, 21, 1, 1, Yes], [3, 22, 0, 0, Yes], [3, 23, 24, 24, Yes], [3, 24, 23, 23, Yes], [4, 0, 21, 21, Yes], [4, 1, 20, 20, Yes], [4, 2, 19, 19, Yes], [4, 3, 18, 18, Yes], [4, 4, 17, 17, Yes], [4, 5, 16, 16, Yes], [4, 6, 15, 15, Yes], [4, 7, 14, 14, Yes], [4, 8, 13, 13, Yes], [4, 9, 12, 12, Yes], [4, 10, 11, 11, Yes], [4, 11, 10, 10, Yes], [4, 12, 9, 9, Yes], [4, 13, 8, 8, Yes], [4, 14, 7, 7, Yes], [4, 15, 6, 6, Yes], [4, 16, 5, 5, Yes], [4, 17, 4, 4, Yes], [4, 18, 3, 3, Yes], [4, 19, 2, 2, Yes], [4, 20, 1, 1, Yes], [4, 21, 0, 0, Yes], [4, 22, 24, 24, Yes], [4, 23, 23, 23, Yes], [4, 24, 22, 22, Yes], [5, 0, 20, 20, Yes], [5, 1, 19, 19, Yes], [5, 2, 18, 18, Yes], [5, 3, 17, 17, Yes], [5, 4, 16, 16, Yes], [5, 5, 15, 15, Yes], [5, 6, 14, 14, Yes], [5, 7, 13, 13, Yes], [5, 8, 12, 12, Yes], [5, 9, 11, 11, Yes], [5, 10, 10, 10, Yes], [5, 11, 9, 9, Yes], [5, 12, 8, 8, Yes], [5, 13, 7, 7, Yes], [5, 14, 6, 6, Yes], [5, 15, 5, 5, Yes], [5, 16, 4, 4, Yes], [5, 17, 3, 3, Yes], [5, 18, 2, 2, Yes], [5, 19, 1, 1, Yes], [5, 20, 0, 0, Yes], [5, 21, 24, 24, Yes], [5, 22, 23, 23, Yes], [5, 23, 22, 22, Yes], [5, 24, 21, 21, Yes], [6, 0, 19, 19, Yes], [6, 1, 18, 18, Yes], [6, 2, 17, 17, Yes], [6, 3, 16, 16, Yes], [6, 4, 15, 15, Yes], [6, 5, 14, 14, Yes], [6, 6, 13, 13, Yes], [6, 7, 12, 12, Yes], [6, 8, 11, 11, Yes], [6, 9, 10, 10, Yes], [6, 10, 9, 9, Yes], [6, 11, 8, 8, Yes], [6, 12, 7, 7, Yes], [6, 13, 6, 6, Yes], [6, 14, 5, 5, Yes], [6, 15, 4, 4, Yes], [6, 16, 3, 3, Yes], [6, 17, 2, 2, Yes], [6, 18, 1, 1, Yes], [6, 19, 0, 0, Yes], [6, 20, 24, 24, Yes], [6, 21, 23, 23, Yes], [6, 22, 22, 22, Yes], [6, 23, 21, 21, Yes], [6, 24, 20, 20, Yes], [7, 0, 18, 18, Yes], [7, 1, 17, 17, Yes], [7, 2, 16, 16, Yes], [7, 3, 15, 15, Yes], [7, 4, 14, 14, Yes], [7, 5, 13, 13, Yes], [7, 6, 12, 12, Yes], [7, 7, 11, 11, Yes], [7, 8, 10, 10, Yes], [7, 9, 9, 9, Yes], [7, 10, 8, 8, Yes], [7, 11, 7, 7, Yes], [7, 12, 6, 6, Yes], [7, 13, 5, 5, Yes], [7, 14, 4, 4, Yes], [7, 15, 3, 3, Yes], [7, 16, 2, 2, Yes], [7, 17, 1, 1, Yes], [7, 18, 0, 0, Yes], [7, 19, 24, 24, Yes], [7, 20, 23, 23, Yes], [7, 21, 22, 22, Yes], [7, 22, 21, 21, Yes], [7, 23, 20, 20, Yes], [7, 24, 19, 19, Yes], [8, 0, 17, 17, Yes], [8, 1, 16, 16, Yes], [8, 2, 15, 15, Yes], [8, 3, 14, 14, Yes], [8, 4, 13, 13, Yes], [8, 5, 12, 12, Yes], [8, 6, 11, 11, Yes], [8, 7, 10, 10, Yes], [8, 8, 9, 9, Yes], [8, 9, 8, 8, Yes], [8, 10, 7, 7, Yes], [8, 11, 6, 6, Yes], [8, 12, 5, 5, Yes], [8, 13, 4, 4, Yes], [8, 14, 3, 3, Yes], [8, 15, 2, 2, Yes], [8, 16, 1, 1, Yes], [8, 17, 0, 0, Yes], [8, 18, 24, 24, Yes], [8, 19, 23, 23, Yes], [8, 20, 22, 22, Yes], [8, 21, 21, 21, Yes], [8, 22, 20, 20, Yes], [8, 23, 19, 19, Yes], [8, 24, 18, 18, Yes], [9, 0, 16, 16, Yes], [9, 1, 15, 15, Yes], [9, 2, 14, 14, Yes], [9, 3, 13, 13, Yes], [9, 4, 12, 12, Yes], [9, 5, 11, 11, Yes], [9, 6, 10, 10, Yes], [9, 7, 9, 9, Yes], [9, 8, 8, 8, Yes], [9, 9, 7, 7, Yes], [9, 10, 6, 6, Yes], [9, 11, 5, 5, Yes], [9, 12, 4, 4, Yes], [9, 13, 3, 3, Yes], [9, 14, 2, 2, Yes], [9, 15, 1, 1, Yes], [9, 16, 0, 0, Yes], [9, 17, 24, 24, Yes], [9, 18, 23, 23, Yes], [9, 19, 22, 22, Yes], [9, 20, 21, 21, Yes], [9, 21, 20, 20, Yes], [9, 22, 19, 19, Yes], [9, 23, 18, 18, Yes], [9, 24, 17, 17, Yes], [10, 0, 15, 15, Yes], [10, 1, 14, 14, Yes], [10, 2, 13, 13, Yes], [10, 3, 12, 12, Yes], [10, 4, 11, 11, Yes], [10, 5, 10, 10, Yes], [10, 6, 9, 9, Yes], [10, 7, 8, 8, Yes], [10, 8, 7, 7, Yes], [10, 9, 6, 6, Yes], [10, 10, 5, 5, Yes], [10, 11, 4, 4, Yes], [10, 12, 3, 3, Yes], [10, 13, 2, 2, Yes], [10, 14, 1, 1, Yes], [10, 15, 0, 0, Yes], [10, 16, 24, 24, Yes], [10, 17, 23, 23, Yes], [10, 18, 22, 22, Yes], [10, 19, 21, 21, Yes], [10, 20, 20, 20, Yes], [10, 21, 19, 19, Yes], [10, 22, 18, 18, Yes], [10, 23, 17, 17, Yes], [10, 24, 16, 16, Yes], [11, 0, 14, 14, Yes], [11, 1, 13, 13, Yes], [11, 2, 12, 12, Yes], [11, 3, 11, 11, Yes], [11, 4, 10, 10, Yes], [11, 5, 9, 9, Yes], [11, 6, 8, 8, Yes], [11, 7, 7, 7, Yes], [11, 8, 6, 6, Yes], [11, 9, 5, 5, Yes], [11, 10, 4, 4, Yes], [11, 11, 3, 3, Yes], [11, 12, 2, 2, Yes], [11, 13, 1, 1, Yes], [11, 14, 0, 0, Yes], [11, 15, 24, 24, Yes], [11, 16, 23, 23, Yes], [11, 17, 22, 22, Yes], [11, 18, 21, 21, Yes], [11, 19, 20, 20, Yes], [11, 20, 19, 19, Yes], [11, 21, 18, 18, Yes], [11, 22, 17, 17, Yes], [11, 23, 16, 16, Yes], [11, 24, 15, 15, Yes], [12, 0, 13, 13, Yes], [12, 1, 12, 12, Yes], [12, 2, 11, 11, Yes], [12, 3, 10, 10, Yes], [12, 4, 9, 9, Yes], [12, 5, 8, 8, Yes], [12, 6, 7, 7, Yes], [12, 7, 6, 6, Yes], [12, 8, 5, 5, Yes], [12, 9, 4, 4, Yes], [12, 10, 3, 3, Yes], [12, 11, 2, 2, Yes], [12, 12, 1, 1, Yes], [12, 13, 0, 0, Yes], [12, 14, 24, 24, Yes], [12, 15, 23, 23, Yes], [12, 16, 22, 22, Yes], [12, 17, 21, 21, Yes], [12, 18, 20, 20, Yes], [12, 19, 19, 19, Yes], [12, 20, 18, 18, Yes], [12, 21, 17, 17, Yes], [12, 22, 16, 16, Yes], [12, 23, 15, 15, Yes], [12, 24, 14, 14, Yes], [13, 0, 12, 12, Yes], [13, 1, 11, 11, Yes], [13, 2, 10, 10, Yes], [13, 3, 9, 9, Yes], [13, 4, 8, 8, Yes], [13, 5, 7, 7, Yes], [13, 6, 6, 6, Yes], [13, 7, 5, 5, Yes], [13, 8, 4, 4, Yes], [13, 9, 3, 3, Yes], [13, 10, 2, 2, Yes], [13, 11, 1, 1, Yes], [13, 12, 0, 0, Yes], [13, 13, 24, 24, Yes], [13, 14, 23, 23, Yes], [13, 15, 22, 22, Yes], [13, 16, 21, 21, Yes], [13, 17, 20, 20, Yes], [13, 18, 19, 19, Yes], [13, 19, 18, 18, Yes], [13, 20, 17, 17, Yes], [13, 21, 16, 16, Yes], [13, 22, 15, 15, Yes], [13, 23, 14, 14, Yes], [13, 24, 13, 13, Yes], [14, 0, 11, 11, Yes], [14, 1, 10, 10, Yes], [14, 2, 9, 9, Yes], [14, 3, 8, 8, Yes], [14, 4, 7, 7, Yes], [14, 5, 6, 6, Yes], [14, 6, 5, 5, Yes], [14, 7, 4, 4, Yes], [14, 8, 3, 3, Yes], [14, 9, 2, 2, Yes], [14, 10, 1, 1, Yes], [14, 11, 0, 0, Yes], [14, 12, 24, 24, Yes], [14, 13, 23, 23, Yes], [14, 14, 22, 22, Yes], [14, 15, 21, 21, Yes], [14, 16, 20, 20, Yes], [14, 17, 19, 19, Yes], [14, 18, 18, 18, Yes], [14, 19, 17, 17, Yes], [14, 20, 16, 16, Yes], [14, 21, 15, 15, Yes], [14, 22, 14, 14, Yes], [14, 23, 13, 13, Yes], [14, 24, 12, 12, Yes], [15, 0, 10, 10, Yes], [15, 1, 9, 9, Yes], [15, 2, 8, 8, Yes], [15, 3, 7, 7, Yes], [15, 4, 6, 6, Yes], [15, 5, 5, 5, Yes], [15, 6, 4, 4, Yes], [15, 7, 3, 3, Yes], [15, 8, 2, 2, Yes], [15, 9, 1, 1, Yes], [15, 10, 0, 0, Yes], [15, 11, 24, 24, Yes], [15, 12, 23, 23, Yes], [15, 13, 22, 22, Yes], [15, 14, 21, 21, Yes], [15, 15, 20, 20, Yes], [15, 16, 19, 19, Yes], [15, 17, 18, 18, Yes], [15, 18, 17, 17, Yes], [15, 19, 16, 16, Yes], [15, 20, 15, 15, Yes], [15, 21, 14, 14, Yes], [15, 22, 13, 13, Yes], [15, 23, 12, 12, Yes], [15, 24, 11, 11, Yes], [16, 0, 9, 9, Yes], [16, 1, 8, 8, Yes], [16, 2, 7, 7, Yes], [16, 3, 6, 6, Yes], [16, 4, 5, 5, Yes], [16, 5, 4, 4, Yes], [16, 6, 3, 3, Yes], [16, 7, 2, 2, Yes], [16, 8, 1, 1, Yes], [16, 9, 0, 0, Yes], [16, 10, 24, 24, Yes], [16, 11, 23, 23, Yes], [16, 12, 22, 22, Yes], [16, 13, 21, 21, Yes], [16, 14, 20, 20, Yes], [16, 15, 19, 19, Yes], [16, 16, 18, 18, Yes], [16, 17, 17, 17, Yes], [16, 18, 16, 16, Yes], [16, 19, 15, 15, Yes], [16, 20, 14, 14, Yes], [16, 21, 13, 13, Yes], [16, 22, 12, 12, Yes], [16, 23, 11, 11, Yes], [16, 24, 10, 10, Yes], [17, 0, 8, 8, Yes], [17, 1, 7, 7, Yes], [17, 2, 6, 6, Yes], [17, 3, 5, 5, Yes], [17, 4, 4, 4, Yes], [17, 5, 3, 3, Yes], [17, 6, 2, 2, Yes], [17, 7, 1, 1, Yes], [17, 8, 0, 0, Yes], [17, 9, 24, 24, Yes], [17, 10, 23, 23, Yes], [17, 11, 22, 22, Yes], [17, 12, 21, 21, Yes], [17, 13, 20, 20, Yes], [17, 14, 19, 19, Yes], [17, 15, 18, 18, Yes], [17, 16, 17, 17, Yes], [17, 17, 16, 16, Yes], [17, 18, 15, 15, Yes], [17, 19, 14, 14, Yes], [17, 20, 13, 13, Yes], [17, 21, 12, 12, Yes], [17, 22, 11, 11, Yes], [17, 23, 10, 10, Yes], [17, 24, 9, 9, Yes], [18, 0, 7, 7, Yes], [18, 1, 6, 6, Yes], [18, 2, 5, 5, Yes], [18, 3, 4, 4, Yes], [18, 4, 3, 3, Yes], [18, 5, 2, 2, Yes], [18, 6, 1, 1, Yes], [18, 7, 0, 0, Yes], [18, 8, 24, 24, Yes], [18, 9, 23, 23, Yes], [18, 10, 22, 22, Yes], [18, 11, 21, 21, Yes], [18, 12, 20, 20, Yes], [18, 13, 19, 19, Yes], [18, 14, 18, 18, Yes], [18, 15, 17, 17, Yes], [18, 16, 16, 16, Yes], [18, 17, 15, 15, Yes], [18, 18, 14, 14, Yes], [18, 19, 13, 13, Yes], [18, 20, 12, 12, Yes], [18, 21, 11, 11, Yes], [18, 22, 10, 10, Yes], [18, 23, 9, 9, Yes], [18, 24, 8, 8, Yes], [19, 0, 6, 6, Yes], [19, 1, 5, 5, Yes], [19, 2, 4, 4, Yes], [19, 3, 3, 3, Yes], [19, 4, 2, 2, Yes], [19, 5, 1, 1, Yes], [19, 6, 0, 0, Yes], [19, 7, 24, 24, Yes], [19, 8, 23, 23, Yes], [19, 9, 22, 22, Yes], [19, 10, 21, 21, Yes], [19, 11, 20, 20, Yes], [19, 12, 19, 19, Yes], [19, 13, 18, 18, Yes], [19, 14, 17, 17, Yes], [19, 15, 16, 16, Yes], [19, 16, 15, 15, Yes], [19, 17, 14, 14, Yes], [19, 18, 13, 13, Yes], [19, 19, 12, 12, Yes], [19, 20, 11, 11, Yes], [19, 21, 10, 10, Yes], [19, 22, 9, 9, Yes], [19, 23, 8, 8, Yes], [19, 24, 7, 7, Yes], [20, 0, 5, 5, Yes], [20, 1, 4, 4, Yes], [20, 2, 3, 3, Yes], [20, 3, 2, 2, Yes], [20, 4, 1, 1, Yes], [20, 5, 0, 0, Yes], [20, 6, 24, 24, Yes], [20, 7, 23, 23, Yes], [20, 8, 22, 22, Yes], [20, 9, 21, 21, Yes], [20, 10, 20, 20, Yes], [20, 11, 19, 19, Yes], [20, 12, 18, 18, Yes], [20, 13, 17, 17, Yes], [20, 14, 16, 16, Yes], [20, 15, 15, 15, Yes], [20, 16, 14, 14, Yes], [20, 17, 13, 13, Yes], [20, 18, 12, 12, Yes], [20, 19, 11, 11, Yes], [20, 20, 10, 10, Yes], [20, 21, 9, 9, Yes], [20, 22, 8, 8, Yes], [20, 23, 7, 7, Yes], [20, 24, 6, 6, Yes], [21, 0, 4, 4, Yes], [21, 1, 3, 3, Yes], [21, 2, 2, 2, Yes], [21, 3, 1, 1, Yes], [21, 4, 0, 0, Yes], [21, 5, 24, 24, Yes], [21, 6, 23, 23, Yes], [21, 7, 22, 22, Yes], [21, 8, 21, 21, Yes], [21, 9, 20, 20, Yes], [21, 10, 19, 19, Yes], [21, 11, 18, 18, Yes], [21, 12, 17, 17, Yes], [21, 13, 16, 16, Yes], [21, 14, 15, 15, Yes], [21, 15, 14, 14, Yes], [21, 16, 13, 13, Yes], [21, 17, 12, 12, Yes], [21, 18, 11, 11, Yes], [21, 19, 10, 10, Yes], [21, 20, 9, 9, Yes], [21, 21, 8, 8, Yes], [21, 22, 7, 7, Yes], [21, 23, 6, 6, Yes], [21, 24, 5, 5, Yes], [22, 0, 3, 3, Yes], [22, 1, 2, 2, Yes], [22, 2, 1, 1, Yes], [22, 3, 0, 0, Yes], [22, 4, 24, 24, Yes], [22, 5, 23, 23, Yes], [22, 6, 22, 22, Yes], [22, 7, 21, 21, Yes], [22, 8, 20, 20, Yes], [22, 9, 19, 19, Yes], [22, 10, 18, 18, Yes], [22, 11, 17, 17, Yes], [22, 12, 16, 16, Yes], [22, 13, 15, 15, Yes], [22, 14, 14, 14, Yes], [22, 15, 13, 13, Yes], [22, 16, 12, 12, Yes], [22, 17, 11, 11, Yes], [22, 18, 10, 10, Yes], [22, 19, 9, 9, Yes], [22, 20, 8, 8, Yes], [22, 21, 7, 7, Yes], [22, 22, 6, 6, Yes], [22, 23, 5, 5, Yes], [22, 24, 4, 4, Yes], [23, 0, 2, 2, Yes], [23, 1, 1, 1, Yes], [23, 2, 0, 0, Yes], [23, 3, 24, 24, Yes], [23, 4, 23, 23, Yes], [23, 5, 22, 22, Yes], [23, 6, 21, 21, Yes], [23, 7, 20, 20, Yes], [23, 8, 19, 19, Yes], [23, 9, 18, 18, Yes], [23, 10, 17, 17, Yes], [23, 11, 16, 16, Yes], [23, 12, 15, 15, Yes], [23, 13, 14, 14, Yes], [23, 14, 13, 13, Yes], [23, 15, 12, 12, Yes], [23, 16, 11, 11, Yes], [23, 17, 10, 10, Yes], [23, 18, 9, 9, Yes], [23, 19, 8, 8, Yes], [23, 20, 7, 7, Yes], [23, 21, 6, 6, Yes], [23, 22, 5, 5, Yes], [23, 23, 4, 4, Yes], [23, 24, 3, 3, Yes], [24, 0, 1, 1, Yes], [24, 1, 0, 0, Yes], [24, 2, 24, 24, Yes], [24, 3, 23, 23, Yes], [24, 4, 22, 22, Yes], [24, 5, 21, 21, Yes], [24, 6, 20, 20, Yes], [24, 7, 19, 19, Yes], [24, 8, 18, 18, Yes], [24, 9, 17, 17, Yes], [24, 10, 16, 16, Yes], [24, 11, 15, 15, Yes], [24, 12, 14, 14, Yes], [24, 13, 13, 13, Yes], [24, 14, 12, 12, Yes], [24, 15, 11, 11, Yes], [24, 16, 10, 10, Yes], [24, 17, 9, 9, Yes], [24, 18, 8, 8, Yes], [24, 19, 7, 7, Yes], [24, 20, 6, 6, Yes], [24, 21, 5, 5, Yes], [24, 22, 4, 4, Yes], [24, 23, 3, 3, Yes], [24, 24, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-homomorphism-qualification · candidate
24. Qualify Both Groups and Every Product Pair before Applying Kernel Theorems
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]], target identity=0, and total mapping=[0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).
View answer and explanation
(26, 26, [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 25, 25, Yes], [0, 2, 24, 24, Yes], [0, 3, 23, 23, Yes], [0, 4, 22, 22, Yes], [0, 5, 21, 21, Yes], [0, 6, 20, 20, Yes], [0, 7, 19, 19, Yes], [0, 8, 18, 18, Yes], [0, 9, 17, 17, Yes], [0, 10, 16, 16, Yes], [0, 11, 15, 15, Yes], [0, 12, 14, 14, Yes], [0, 13, 13, 13, Yes], [0, 14, 12, 12, Yes], [0, 15, 11, 11, Yes], [0, 16, 10, 10, Yes], [0, 17, 9, 9, Yes], [0, 18, 8, 8, Yes], [0, 19, 7, 7, Yes], [0, 20, 6, 6, Yes], [0, 21, 5, 5, Yes], [0, 22, 4, 4, Yes], [0, 23, 3, 3, Yes], [0, 24, 2, 2, Yes], [0, 25, 1, 1, Yes], [1, 0, 25, 25, Yes], [1, 1, 24, 24, Yes], [1, 2, 23, 23, Yes], [1, 3, 22, 22, Yes], [1, 4, 21, 21, Yes], [1, 5, 20, 20, Yes], [1, 6, 19, 19, Yes], [1, 7, 18, 18, Yes], [1, 8, 17, 17, Yes], [1, 9, 16, 16, Yes], [1, 10, 15, 15, Yes], [1, 11, 14, 14, Yes], [1, 12, 13, 13, Yes], [1, 13, 12, 12, Yes], [1, 14, 11, 11, Yes], [1, 15, 10, 10, Yes], [1, 16, 9, 9, Yes], [1, 17, 8, 8, Yes], [1, 18, 7, 7, Yes], [1, 19, 6, 6, Yes], [1, 20, 5, 5, Yes], [1, 21, 4, 4, Yes], [1, 22, 3, 3, Yes], [1, 23, 2, 2, Yes], [1, 24, 1, 1, Yes], [1, 25, 0, 0, Yes], [2, 0, 24, 24, Yes], [2, 1, 23, 23, Yes], [2, 2, 22, 22, Yes], [2, 3, 21, 21, Yes], [2, 4, 20, 20, Yes], [2, 5, 19, 19, Yes], [2, 6, 18, 18, Yes], [2, 7, 17, 17, Yes], [2, 8, 16, 16, Yes], [2, 9, 15, 15, Yes], [2, 10, 14, 14, Yes], [2, 11, 13, 13, Yes], [2, 12, 12, 12, Yes], [2, 13, 11, 11, Yes], [2, 14, 10, 10, Yes], [2, 15, 9, 9, Yes], [2, 16, 8, 8, Yes], [2, 17, 7, 7, Yes], [2, 18, 6, 6, Yes], [2, 19, 5, 5, Yes], [2, 20, 4, 4, Yes], [2, 21, 3, 3, Yes], [2, 22, 2, 2, Yes], [2, 23, 1, 1, Yes], [2, 24, 0, 0, Yes], [2, 25, 25, 25, Yes], [3, 0, 23, 23, Yes], [3, 1, 22, 22, Yes], [3, 2, 21, 21, Yes], [3, 3, 20, 20, Yes], [3, 4, 19, 19, Yes], [3, 5, 18, 18, Yes], [3, 6, 17, 17, Yes], [3, 7, 16, 16, Yes], [3, 8, 15, 15, Yes], [3, 9, 14, 14, Yes], [3, 10, 13, 13, Yes], [3, 11, 12, 12, Yes], [3, 12, 11, 11, Yes], [3, 13, 10, 10, Yes], [3, 14, 9, 9, Yes], [3, 15, 8, 8, Yes], [3, 16, 7, 7, Yes], [3, 17, 6, 6, Yes], [3, 18, 5, 5, Yes], [3, 19, 4, 4, Yes], [3, 20, 3, 3, Yes], [3, 21, 2, 2, Yes], [3, 22, 1, 1, Yes], [3, 23, 0, 0, Yes], [3, 24, 25, 25, Yes], [3, 25, 24, 24, Yes], [4, 0, 22, 22, Yes], [4, 1, 21, 21, Yes], [4, 2, 20, 20, Yes], [4, 3, 19, 19, Yes], [4, 4, 18, 18, Yes], [4, 5, 17, 17, Yes], [4, 6, 16, 16, Yes], [4, 7, 15, 15, Yes], [4, 8, 14, 14, Yes], [4, 9, 13, 13, Yes], [4, 10, 12, 12, Yes], [4, 11, 11, 11, Yes], [4, 12, 10, 10, Yes], [4, 13, 9, 9, Yes], [4, 14, 8, 8, Yes], [4, 15, 7, 7, Yes], [4, 16, 6, 6, Yes], [4, 17, 5, 5, Yes], [4, 18, 4, 4, Yes], [4, 19, 3, 3, Yes], [4, 20, 2, 2, Yes], [4, 21, 1, 1, Yes], [4, 22, 0, 0, Yes], [4, 23, 25, 25, Yes], [4, 24, 24, 24, Yes], [4, 25, 23, 23, Yes], [5, 0, 21, 21, Yes], [5, 1, 20, 20, Yes], [5, 2, 19, 19, Yes], [5, 3, 18, 18, Yes], [5, 4, 17, 17, Yes], [5, 5, 16, 16, Yes], [5, 6, 15, 15, Yes], [5, 7, 14, 14, Yes], [5, 8, 13, 13, Yes], [5, 9, 12, 12, Yes], [5, 10, 11, 11, Yes], [5, 11, 10, 10, Yes], [5, 12, 9, 9, Yes], [5, 13, 8, 8, Yes], [5, 14, 7, 7, Yes], [5, 15, 6, 6, Yes], [5, 16, 5, 5, Yes], [5, 17, 4, 4, Yes], [5, 18, 3, 3, Yes], [5, 19, 2, 2, Yes], [5, 20, 1, 1, Yes], [5, 21, 0, 0, Yes], [5, 22, 25, 25, Yes], [5, 23, 24, 24, Yes], [5, 24, 23, 23, Yes], [5, 25, 22, 22, Yes], [6, 0, 20, 20, Yes], [6, 1, 19, 19, Yes], [6, 2, 18, 18, Yes], [6, 3, 17, 17, Yes], [6, 4, 16, 16, Yes], [6, 5, 15, 15, Yes], [6, 6, 14, 14, Yes], [6, 7, 13, 13, Yes], [6, 8, 12, 12, Yes], [6, 9, 11, 11, Yes], [6, 10, 10, 10, Yes], [6, 11, 9, 9, Yes], [6, 12, 8, 8, Yes], [6, 13, 7, 7, Yes], [6, 14, 6, 6, Yes], [6, 15, 5, 5, Yes], [6, 16, 4, 4, Yes], [6, 17, 3, 3, Yes], [6, 18, 2, 2, Yes], [6, 19, 1, 1, Yes], [6, 20, 0, 0, Yes], [6, 21, 25, 25, Yes], [6, 22, 24, 24, Yes], [6, 23, 23, 23, Yes], [6, 24, 22, 22, Yes], [6, 25, 21, 21, Yes], [7, 0, 19, 19, Yes], [7, 1, 18, 18, Yes], [7, 2, 17, 17, Yes], [7, 3, 16, 16, Yes], [7, 4, 15, 15, Yes], [7, 5, 14, 14, Yes], [7, 6, 13, 13, Yes], [7, 7, 12, 12, Yes], [7, 8, 11, 11, Yes], [7, 9, 10, 10, Yes], [7, 10, 9, 9, Yes], [7, 11, 8, 8, Yes], [7, 12, 7, 7, Yes], [7, 13, 6, 6, Yes], [7, 14, 5, 5, Yes], [7, 15, 4, 4, Yes], [7, 16, 3, 3, Yes], [7, 17, 2, 2, Yes], [7, 18, 1, 1, Yes], [7, 19, 0, 0, Yes], [7, 20, 25, 25, Yes], [7, 21, 24, 24, Yes], [7, 22, 23, 23, Yes], [7, 23, 22, 22, Yes], [7, 24, 21, 21, Yes], [7, 25, 20, 20, Yes], [8, 0, 18, 18, Yes], [8, 1, 17, 17, Yes], [8, 2, 16, 16, Yes], [8, 3, 15, 15, Yes], [8, 4, 14, 14, Yes], [8, 5, 13, 13, Yes], [8, 6, 12, 12, Yes], [8, 7, 11, 11, Yes], [8, 8, 10, 10, Yes], [8, 9, 9, 9, Yes], [8, 10, 8, 8, Yes], [8, 11, 7, 7, Yes], [8, 12, 6, 6, Yes], [8, 13, 5, 5, Yes], [8, 14, 4, 4, Yes], [8, 15, 3, 3, Yes], [8, 16, 2, 2, Yes], [8, 17, 1, 1, Yes], [8, 18, 0, 0, Yes], [8, 19, 25, 25, Yes], [8, 20, 24, 24, Yes], [8, 21, 23, 23, Yes], [8, 22, 22, 22, Yes], [8, 23, 21, 21, Yes], [8, 24, 20, 20, Yes], [8, 25, 19, 19, Yes], [9, 0, 17, 17, Yes], [9, 1, 16, 16, Yes], [9, 2, 15, 15, Yes], [9, 3, 14, 14, Yes], [9, 4, 13, 13, Yes], [9, 5, 12, 12, Yes], [9, 6, 11, 11, Yes], [9, 7, 10, 10, Yes], [9, 8, 9, 9, Yes], [9, 9, 8, 8, Yes], [9, 10, 7, 7, Yes], [9, 11, 6, 6, Yes], [9, 12, 5, 5, Yes], [9, 13, 4, 4, Yes], [9, 14, 3, 3, Yes], [9, 15, 2, 2, Yes], [9, 16, 1, 1, Yes], [9, 17, 0, 0, Yes], [9, 18, 25, 25, Yes], [9, 19, 24, 24, Yes], [9, 20, 23, 23, Yes], [9, 21, 22, 22, Yes], [9, 22, 21, 21, Yes], [9, 23, 20, 20, Yes], [9, 24, 19, 19, Yes], [9, 25, 18, 18, Yes], [10, 0, 16, 16, Yes], [10, 1, 15, 15, Yes], [10, 2, 14, 14, Yes], [10, 3, 13, 13, Yes], [10, 4, 12, 12, Yes], [10, 5, 11, 11, Yes], [10, 6, 10, 10, Yes], [10, 7, 9, 9, Yes], [10, 8, 8, 8, Yes], [10, 9, 7, 7, Yes], [10, 10, 6, 6, Yes], [10, 11, 5, 5, Yes], [10, 12, 4, 4, Yes], [10, 13, 3, 3, Yes], [10, 14, 2, 2, Yes], [10, 15, 1, 1, Yes], [10, 16, 0, 0, Yes], [10, 17, 25, 25, Yes], [10, 18, 24, 24, Yes], [10, 19, 23, 23, Yes], [10, 20, 22, 22, Yes], [10, 21, 21, 21, Yes], [10, 22, 20, 20, Yes], [10, 23, 19, 19, Yes], [10, 24, 18, 18, Yes], [10, 25, 17, 17, Yes], [11, 0, 15, 15, Yes], [11, 1, 14, 14, Yes], [11, 2, 13, 13, Yes], [11, 3, 12, 12, Yes], [11, 4, 11, 11, Yes], [11, 5, 10, 10, Yes], [11, 6, 9, 9, Yes], [11, 7, 8, 8, Yes], [11, 8, 7, 7, Yes], [11, 9, 6, 6, Yes], [11, 10, 5, 5, Yes], [11, 11, 4, 4, Yes], [11, 12, 3, 3, Yes], [11, 13, 2, 2, Yes], [11, 14, 1, 1, Yes], [11, 15, 0, 0, Yes], [11, 16, 25, 25, Yes], [11, 17, 24, 24, Yes], [11, 18, 23, 23, Yes], [11, 19, 22, 22, Yes], [11, 20, 21, 21, Yes], [11, 21, 20, 20, Yes], [11, 22, 19, 19, Yes], [11, 23, 18, 18, Yes], [11, 24, 17, 17, Yes], [11, 25, 16, 16, Yes], [12, 0, 14, 14, Yes], [12, 1, 13, 13, Yes], [12, 2, 12, 12, Yes], [12, 3, 11, 11, Yes], [12, 4, 10, 10, Yes], [12, 5, 9, 9, Yes], [12, 6, 8, 8, Yes], [12, 7, 7, 7, Yes], [12, 8, 6, 6, Yes], [12, 9, 5, 5, Yes], [12, 10, 4, 4, Yes], [12, 11, 3, 3, Yes], [12, 12, 2, 2, Yes], [12, 13, 1, 1, Yes], [12, 14, 0, 0, Yes], [12, 15, 25, 25, Yes], [12, 16, 24, 24, Yes], [12, 17, 23, 23, Yes], [12, 18, 22, 22, Yes], [12, 19, 21, 21, Yes], [12, 20, 20, 20, Yes], [12, 21, 19, 19, Yes], [12, 22, 18, 18, Yes], [12, 23, 17, 17, Yes], [12, 24, 16, 16, Yes], [12, 25, 15, 15, Yes], [13, 0, 13, 13, Yes], [13, 1, 12, 12, Yes], [13, 2, 11, 11, Yes], [13, 3, 10, 10, Yes], [13, 4, 9, 9, Yes], [13, 5, 8, 8, Yes], [13, 6, 7, 7, Yes], [13, 7, 6, 6, Yes], [13, 8, 5, 5, Yes], [13, 9, 4, 4, Yes], [13, 10, 3, 3, Yes], [13, 11, 2, 2, Yes], [13, 12, 1, 1, Yes], [13, 13, 0, 0, Yes], [13, 14, 25, 25, Yes], [13, 15, 24, 24, Yes], [13, 16, 23, 23, Yes], [13, 17, 22, 22, Yes], [13, 18, 21, 21, Yes], [13, 19, 20, 20, Yes], [13, 20, 19, 19, Yes], [13, 21, 18, 18, Yes], [13, 22, 17, 17, Yes], [13, 23, 16, 16, Yes], [13, 24, 15, 15, Yes], [13, 25, 14, 14, Yes], [14, 0, 12, 12, Yes], [14, 1, 11, 11, Yes], [14, 2, 10, 10, Yes], [14, 3, 9, 9, Yes], [14, 4, 8, 8, Yes], [14, 5, 7, 7, Yes], [14, 6, 6, 6, Yes], [14, 7, 5, 5, Yes], [14, 8, 4, 4, Yes], [14, 9, 3, 3, Yes], [14, 10, 2, 2, Yes], [14, 11, 1, 1, Yes], [14, 12, 0, 0, Yes], [14, 13, 25, 25, Yes], [14, 14, 24, 24, Yes], [14, 15, 23, 23, Yes], [14, 16, 22, 22, Yes], [14, 17, 21, 21, Yes], [14, 18, 20, 20, Yes], [14, 19, 19, 19, Yes], [14, 20, 18, 18, Yes], [14, 21, 17, 17, Yes], [14, 22, 16, 16, Yes], [14, 23, 15, 15, Yes], [14, 24, 14, 14, Yes], [14, 25, 13, 13, Yes], [15, 0, 11, 11, Yes], [15, 1, 10, 10, Yes], [15, 2, 9, 9, Yes], [15, 3, 8, 8, Yes], [15, 4, 7, 7, Yes], [15, 5, 6, 6, Yes], [15, 6, 5, 5, Yes], [15, 7, 4, 4, Yes], [15, 8, 3, 3, Yes], [15, 9, 2, 2, Yes], [15, 10, 1, 1, Yes], [15, 11, 0, 0, Yes], [15, 12, 25, 25, Yes], [15, 13, 24, 24, Yes], [15, 14, 23, 23, Yes], [15, 15, 22, 22, Yes], [15, 16, 21, 21, Yes], [15, 17, 20, 20, Yes], [15, 18, 19, 19, Yes], [15, 19, 18, 18, Yes], [15, 20, 17, 17, Yes], [15, 21, 16, 16, Yes], [15, 22, 15, 15, Yes], [15, 23, 14, 14, Yes], [15, 24, 13, 13, Yes], [15, 25, 12, 12, Yes], [16, 0, 10, 10, Yes], [16, 1, 9, 9, Yes], [16, 2, 8, 8, Yes], [16, 3, 7, 7, Yes], [16, 4, 6, 6, Yes], [16, 5, 5, 5, Yes], [16, 6, 4, 4, Yes], [16, 7, 3, 3, Yes], [16, 8, 2, 2, Yes], [16, 9, 1, 1, Yes], [16, 10, 0, 0, Yes], [16, 11, 25, 25, Yes], [16, 12, 24, 24, Yes], [16, 13, 23, 23, Yes], [16, 14, 22, 22, Yes], [16, 15, 21, 21, Yes], [16, 16, 20, 20, Yes], [16, 17, 19, 19, Yes], [16, 18, 18, 18, Yes], [16, 19, 17, 17, Yes], [16, 20, 16, 16, Yes], [16, 21, 15, 15, Yes], [16, 22, 14, 14, Yes], [16, 23, 13, 13, Yes], [16, 24, 12, 12, Yes], [16, 25, 11, 11, Yes], [17, 0, 9, 9, Yes], [17, 1, 8, 8, Yes], [17, 2, 7, 7, Yes], [17, 3, 6, 6, Yes], [17, 4, 5, 5, Yes], [17, 5, 4, 4, Yes], [17, 6, 3, 3, Yes], [17, 7, 2, 2, Yes], [17, 8, 1, 1, Yes], [17, 9, 0, 0, Yes], [17, 10, 25, 25, Yes], [17, 11, 24, 24, Yes], [17, 12, 23, 23, Yes], [17, 13, 22, 22, Yes], [17, 14, 21, 21, Yes], [17, 15, 20, 20, Yes], [17, 16, 19, 19, Yes], [17, 17, 18, 18, Yes], [17, 18, 17, 17, Yes], [17, 19, 16, 16, Yes], [17, 20, 15, 15, Yes], [17, 21, 14, 14, Yes], [17, 22, 13, 13, Yes], [17, 23, 12, 12, Yes], [17, 24, 11, 11, Yes], [17, 25, 10, 10, Yes], [18, 0, 8, 8, Yes], [18, 1, 7, 7, Yes], [18, 2, 6, 6, Yes], [18, 3, 5, 5, Yes], [18, 4, 4, 4, Yes], [18, 5, 3, 3, Yes], [18, 6, 2, 2, Yes], [18, 7, 1, 1, Yes], [18, 8, 0, 0, Yes], [18, 9, 25, 25, Yes], [18, 10, 24, 24, Yes], [18, 11, 23, 23, Yes], [18, 12, 22, 22, Yes], [18, 13, 21, 21, Yes], [18, 14, 20, 20, Yes], [18, 15, 19, 19, Yes], [18, 16, 18, 18, Yes], [18, 17, 17, 17, Yes], [18, 18, 16, 16, Yes], [18, 19, 15, 15, Yes], [18, 20, 14, 14, Yes], [18, 21, 13, 13, Yes], [18, 22, 12, 12, Yes], [18, 23, 11, 11, Yes], [18, 24, 10, 10, Yes], [18, 25, 9, 9, Yes], [19, 0, 7, 7, Yes], [19, 1, 6, 6, Yes], [19, 2, 5, 5, Yes], [19, 3, 4, 4, Yes], [19, 4, 3, 3, Yes], [19, 5, 2, 2, Yes], [19, 6, 1, 1, Yes], [19, 7, 0, 0, Yes], [19, 8, 25, 25, Yes], [19, 9, 24, 24, Yes], [19, 10, 23, 23, Yes], [19, 11, 22, 22, Yes], [19, 12, 21, 21, Yes], [19, 13, 20, 20, Yes], [19, 14, 19, 19, Yes], [19, 15, 18, 18, Yes], [19, 16, 17, 17, Yes], [19, 17, 16, 16, Yes], [19, 18, 15, 15, Yes], [19, 19, 14, 14, Yes], [19, 20, 13, 13, Yes], [19, 21, 12, 12, Yes], [19, 22, 11, 11, Yes], [19, 23, 10, 10, Yes], [19, 24, 9, 9, Yes], [19, 25, 8, 8, Yes], [20, 0, 6, 6, Yes], [20, 1, 5, 5, Yes], [20, 2, 4, 4, Yes], [20, 3, 3, 3, Yes], [20, 4, 2, 2, Yes], [20, 5, 1, 1, Yes], [20, 6, 0, 0, Yes], [20, 7, 25, 25, Yes], [20, 8, 24, 24, Yes], [20, 9, 23, 23, Yes], [20, 10, 22, 22, Yes], [20, 11, 21, 21, Yes], [20, 12, 20, 20, Yes], [20, 13, 19, 19, Yes], [20, 14, 18, 18, Yes], [20, 15, 17, 17, Yes], [20, 16, 16, 16, Yes], [20, 17, 15, 15, Yes], [20, 18, 14, 14, Yes], [20, 19, 13, 13, Yes], [20, 20, 12, 12, Yes], [20, 21, 11, 11, Yes], [20, 22, 10, 10, Yes], [20, 23, 9, 9, Yes], [20, 24, 8, 8, Yes], [20, 25, 7, 7, Yes], [21, 0, 5, 5, Yes], [21, 1, 4, 4, Yes], [21, 2, 3, 3, Yes], [21, 3, 2, 2, Yes], [21, 4, 1, 1, Yes], [21, 5, 0, 0, Yes], [21, 6, 25, 25, Yes], [21, 7, 24, 24, Yes], [21, 8, 23, 23, Yes], [21, 9, 22, 22, Yes], [21, 10, 21, 21, Yes], [21, 11, 20, 20, Yes], [21, 12, 19, 19, Yes], [21, 13, 18, 18, Yes], [21, 14, 17, 17, Yes], [21, 15, 16, 16, Yes], [21, 16, 15, 15, Yes], [21, 17, 14, 14, Yes], [21, 18, 13, 13, Yes], [21, 19, 12, 12, Yes], [21, 20, 11, 11, Yes], [21, 21, 10, 10, Yes], [21, 22, 9, 9, Yes], [21, 23, 8, 8, Yes], [21, 24, 7, 7, Yes], [21, 25, 6, 6, Yes], [22, 0, 4, 4, Yes], [22, 1, 3, 3, Yes], [22, 2, 2, 2, Yes], [22, 3, 1, 1, Yes], [22, 4, 0, 0, Yes], [22, 5, 25, 25, Yes], [22, 6, 24, 24, Yes], [22, 7, 23, 23, Yes], [22, 8, 22, 22, Yes], [22, 9, 21, 21, Yes], [22, 10, 20, 20, Yes], [22, 11, 19, 19, Yes], [22, 12, 18, 18, Yes], [22, 13, 17, 17, Yes], [22, 14, 16, 16, Yes], [22, 15, 15, 15, Yes], [22, 16, 14, 14, Yes], [22, 17, 13, 13, Yes], [22, 18, 12, 12, Yes], [22, 19, 11, 11, Yes], [22, 20, 10, 10, Yes], [22, 21, 9, 9, Yes], [22, 22, 8, 8, Yes], [22, 23, 7, 7, Yes], [22, 24, 6, 6, Yes], [22, 25, 5, 5, Yes], [23, 0, 3, 3, Yes], [23, 1, 2, 2, Yes], [23, 2, 1, 1, Yes], [23, 3, 0, 0, Yes], [23, 4, 25, 25, Yes], [23, 5, 24, 24, Yes], [23, 6, 23, 23, Yes], [23, 7, 22, 22, Yes], [23, 8, 21, 21, Yes], [23, 9, 20, 20, Yes], [23, 10, 19, 19, Yes], [23, 11, 18, 18, Yes], [23, 12, 17, 17, Yes], [23, 13, 16, 16, Yes], [23, 14, 15, 15, Yes], [23, 15, 14, 14, Yes], [23, 16, 13, 13, Yes], [23, 17, 12, 12, Yes], [23, 18, 11, 11, Yes], [23, 19, 10, 10, Yes], [23, 20, 9, 9, Yes], [23, 21, 8, 8, Yes], [23, 22, 7, 7, Yes], [23, 23, 6, 6, Yes], [23, 24, 5, 5, Yes], [23, 25, 4, 4, Yes], [24, 0, 2, 2, Yes], [24, 1, 1, 1, Yes], [24, 2, 0, 0, Yes], [24, 3, 25, 25, Yes], [24, 4, 24, 24, Yes], [24, 5, 23, 23, Yes], [24, 6, 22, 22, Yes], [24, 7, 21, 21, Yes], [24, 8, 20, 20, Yes], [24, 9, 19, 19, Yes], [24, 10, 18, 18, Yes], [24, 11, 17, 17, Yes], [24, 12, 16, 16, Yes], [24, 13, 15, 15, Yes], [24, 14, 14, 14, Yes], [24, 15, 13, 13, Yes], [24, 16, 12, 12, Yes], [24, 17, 11, 11, Yes], [24, 18, 10, 10, Yes], [24, 19, 9, 9, Yes], [24, 20, 8, 8, Yes], [24, 21, 7, 7, Yes], [24, 22, 6, 6, Yes], [24, 23, 5, 5, Yes], [24, 24, 4, 4, Yes], [24, 25, 3, 3, Yes], [25, 0, 1, 1, Yes], [25, 1, 0, 0, Yes], [25, 2, 25, 25, Yes], [25, 3, 24, 24, Yes], [25, 4, 23, 23, Yes], [25, 5, 22, 22, Yes], [25, 6, 21, 21, Yes], [25, 7, 20, 20, Yes], [25, 8, 19, 19, Yes], [25, 9, 18, 18, Yes], [25, 10, 17, 17, Yes], [25, 11, 16, 16, Yes], [25, 12, 15, 15, Yes], [25, 13, 14, 14, Yes], [25, 14, 13, 13, Yes], [25, 15, 12, 12, Yes], [25, 16, 11, 11, Yes], [25, 17, 10, 10, Yes], [25, 18, 9, 9, Yes], [25, 19, 8, 8, Yes], [25, 20, 7, 7, Yes], [25, 21, 6, 6, Yes], [25, 22, 5, 5, Yes], [25, 23, 4, 4, Yes], [25, 24, 3, 3, Yes], [25, 25, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)
List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (26, 26, [0, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 25, 25, Yes], [0, 2, 24, 24, Yes], [0, 3, 23, 23, Yes], [0, 4, 22, 22, Yes], [0, 5, 21, 21, Yes], [0, 6, 20, 20, Yes], [0, 7, 19, 19, Yes], [0, 8, 18, 18, Yes], [0, 9, 17, 17, Yes], [0, 10, 16, 16, Yes], [0, 11, 15, 15, Yes], [0, 12, 14, 14, Yes], [0, 13, 13, 13, Yes], [0, 14, 12, 12, Yes], [0, 15, 11, 11, Yes], [0, 16, 10, 10, Yes], [0, 17, 9, 9, Yes], [0, 18, 8, 8, Yes], [0, 19, 7, 7, Yes], [0, 20, 6, 6, Yes], [0, 21, 5, 5, Yes], [0, 22, 4, 4, Yes], [0, 23, 3, 3, Yes], [0, 24, 2, 2, Yes], [0, 25, 1, 1, Yes], [1, 0, 25, 25, Yes], [1, 1, 24, 24, Yes], [1, 2, 23, 23, Yes], [1, 3, 22, 22, Yes], [1, 4, 21, 21, Yes], [1, 5, 20, 20, Yes], [1, 6, 19, 19, Yes], [1, 7, 18, 18, Yes], [1, 8, 17, 17, Yes], [1, 9, 16, 16, Yes], [1, 10, 15, 15, Yes], [1, 11, 14, 14, Yes], [1, 12, 13, 13, Yes], [1, 13, 12, 12, Yes], [1, 14, 11, 11, Yes], [1, 15, 10, 10, Yes], [1, 16, 9, 9, Yes], [1, 17, 8, 8, Yes], [1, 18, 7, 7, Yes], [1, 19, 6, 6, Yes], [1, 20, 5, 5, Yes], [1, 21, 4, 4, Yes], [1, 22, 3, 3, Yes], [1, 23, 2, 2, Yes], [1, 24, 1, 1, Yes], [1, 25, 0, 0, Yes], [2, 0, 24, 24, Yes], [2, 1, 23, 23, Yes], [2, 2, 22, 22, Yes], [2, 3, 21, 21, Yes], [2, 4, 20, 20, Yes], [2, 5, 19, 19, Yes], [2, 6, 18, 18, Yes], [2, 7, 17, 17, Yes], [2, 8, 16, 16, Yes], [2, 9, 15, 15, Yes], [2, 10, 14, 14, Yes], [2, 11, 13, 13, Yes], [2, 12, 12, 12, Yes], [2, 13, 11, 11, Yes], [2, 14, 10, 10, Yes], [2, 15, 9, 9, Yes], [2, 16, 8, 8, Yes], [2, 17, 7, 7, Yes], [2, 18, 6, 6, Yes], [2, 19, 5, 5, Yes], [2, 20, 4, 4, Yes], [2, 21, 3, 3, Yes], [2, 22, 2, 2, Yes], [2, 23, 1, 1, Yes], [2, 24, 0, 0, Yes], [2, 25, 25, 25, Yes], [3, 0, 23, 23, Yes], [3, 1, 22, 22, Yes], [3, 2, 21, 21, Yes], [3, 3, 20, 20, Yes], [3, 4, 19, 19, Yes], [3, 5, 18, 18, Yes], [3, 6, 17, 17, Yes], [3, 7, 16, 16, Yes], [3, 8, 15, 15, Yes], [3, 9, 14, 14, Yes], [3, 10, 13, 13, Yes], [3, 11, 12, 12, Yes], [3, 12, 11, 11, Yes], [3, 13, 10, 10, Yes], [3, 14, 9, 9, Yes], [3, 15, 8, 8, Yes], [3, 16, 7, 7, Yes], [3, 17, 6, 6, Yes], [3, 18, 5, 5, Yes], [3, 19, 4, 4, Yes], [3, 20, 3, 3, Yes], [3, 21, 2, 2, Yes], [3, 22, 1, 1, Yes], [3, 23, 0, 0, Yes], [3, 24, 25, 25, Yes], [3, 25, 24, 24, Yes], [4, 0, 22, 22, Yes], [4, 1, 21, 21, Yes], [4, 2, 20, 20, Yes], [4, 3, 19, 19, Yes], [4, 4, 18, 18, Yes], [4, 5, 17, 17, Yes], [4, 6, 16, 16, Yes], [4, 7, 15, 15, Yes], [4, 8, 14, 14, Yes], [4, 9, 13, 13, Yes], [4, 10, 12, 12, Yes], [4, 11, 11, 11, Yes], [4, 12, 10, 10, Yes], [4, 13, 9, 9, Yes], [4, 14, 8, 8, Yes], [4, 15, 7, 7, Yes], [4, 16, 6, 6, Yes], [4, 17, 5, 5, Yes], [4, 18, 4, 4, Yes], [4, 19, 3, 3, Yes], [4, 20, 2, 2, Yes], [4, 21, 1, 1, Yes], [4, 22, 0, 0, Yes], [4, 23, 25, 25, Yes], [4, 24, 24, 24, Yes], [4, 25, 23, 23, Yes], [5, 0, 21, 21, Yes], [5, 1, 20, 20, Yes], [5, 2, 19, 19, Yes], [5, 3, 18, 18, Yes], [5, 4, 17, 17, Yes], [5, 5, 16, 16, Yes], [5, 6, 15, 15, Yes], [5, 7, 14, 14, Yes], [5, 8, 13, 13, Yes], [5, 9, 12, 12, Yes], [5, 10, 11, 11, Yes], [5, 11, 10, 10, Yes], [5, 12, 9, 9, Yes], [5, 13, 8, 8, Yes], [5, 14, 7, 7, Yes], [5, 15, 6, 6, Yes], [5, 16, 5, 5, Yes], [5, 17, 4, 4, Yes], [5, 18, 3, 3, Yes], [5, 19, 2, 2, Yes], [5, 20, 1, 1, Yes], [5, 21, 0, 0, Yes], [5, 22, 25, 25, Yes], [5, 23, 24, 24, Yes], [5, 24, 23, 23, Yes], [5, 25, 22, 22, Yes], [6, 0, 20, 20, Yes], [6, 1, 19, 19, Yes], [6, 2, 18, 18, Yes], [6, 3, 17, 17, Yes], [6, 4, 16, 16, Yes], [6, 5, 15, 15, Yes], [6, 6, 14, 14, Yes], [6, 7, 13, 13, Yes], [6, 8, 12, 12, Yes], [6, 9, 11, 11, Yes], [6, 10, 10, 10, Yes], [6, 11, 9, 9, Yes], [6, 12, 8, 8, Yes], [6, 13, 7, 7, Yes], [6, 14, 6, 6, Yes], [6, 15, 5, 5, Yes], [6, 16, 4, 4, Yes], [6, 17, 3, 3, Yes], [6, 18, 2, 2, Yes], [6, 19, 1, 1, Yes], [6, 20, 0, 0, Yes], [6, 21, 25, 25, Yes], [6, 22, 24, 24, Yes], [6, 23, 23, 23, Yes], [6, 24, 22, 22, Yes], [6, 25, 21, 21, Yes], [7, 0, 19, 19, Yes], [7, 1, 18, 18, Yes], [7, 2, 17, 17, Yes], [7, 3, 16, 16, Yes], [7, 4, 15, 15, Yes], [7, 5, 14, 14, Yes], [7, 6, 13, 13, Yes], [7, 7, 12, 12, Yes], [7, 8, 11, 11, Yes], [7, 9, 10, 10, Yes], [7, 10, 9, 9, Yes], [7, 11, 8, 8, Yes], [7, 12, 7, 7, Yes], [7, 13, 6, 6, Yes], [7, 14, 5, 5, Yes], [7, 15, 4, 4, Yes], [7, 16, 3, 3, Yes], [7, 17, 2, 2, Yes], [7, 18, 1, 1, Yes], [7, 19, 0, 0, Yes], [7, 20, 25, 25, Yes], [7, 21, 24, 24, Yes], [7, 22, 23, 23, Yes], [7, 23, 22, 22, Yes], [7, 24, 21, 21, Yes], [7, 25, 20, 20, Yes], [8, 0, 18, 18, Yes], [8, 1, 17, 17, Yes], [8, 2, 16, 16, Yes], [8, 3, 15, 15, Yes], [8, 4, 14, 14, Yes], [8, 5, 13, 13, Yes], [8, 6, 12, 12, Yes], [8, 7, 11, 11, Yes], [8, 8, 10, 10, Yes], [8, 9, 9, 9, Yes], [8, 10, 8, 8, Yes], [8, 11, 7, 7, Yes], [8, 12, 6, 6, Yes], [8, 13, 5, 5, Yes], [8, 14, 4, 4, Yes], [8, 15, 3, 3, Yes], [8, 16, 2, 2, Yes], [8, 17, 1, 1, Yes], [8, 18, 0, 0, Yes], [8, 19, 25, 25, Yes], [8, 20, 24, 24, Yes], [8, 21, 23, 23, Yes], [8, 22, 22, 22, Yes], [8, 23, 21, 21, Yes], [8, 24, 20, 20, Yes], [8, 25, 19, 19, Yes], [9, 0, 17, 17, Yes], [9, 1, 16, 16, Yes], [9, 2, 15, 15, Yes], [9, 3, 14, 14, Yes], [9, 4, 13, 13, Yes], [9, 5, 12, 12, Yes], [9, 6, 11, 11, Yes], [9, 7, 10, 10, Yes], [9, 8, 9, 9, Yes], [9, 9, 8, 8, Yes], [9, 10, 7, 7, Yes], [9, 11, 6, 6, Yes], [9, 12, 5, 5, Yes], [9, 13, 4, 4, Yes], [9, 14, 3, 3, Yes], [9, 15, 2, 2, Yes], [9, 16, 1, 1, Yes], [9, 17, 0, 0, Yes], [9, 18, 25, 25, Yes], [9, 19, 24, 24, Yes], [9, 20, 23, 23, Yes], [9, 21, 22, 22, Yes], [9, 22, 21, 21, Yes], [9, 23, 20, 20, Yes], [9, 24, 19, 19, Yes], [9, 25, 18, 18, Yes], [10, 0, 16, 16, Yes], [10, 1, 15, 15, Yes], [10, 2, 14, 14, Yes], [10, 3, 13, 13, Yes], [10, 4, 12, 12, Yes], [10, 5, 11, 11, Yes], [10, 6, 10, 10, Yes], [10, 7, 9, 9, Yes], [10, 8, 8, 8, Yes], [10, 9, 7, 7, Yes], [10, 10, 6, 6, Yes], [10, 11, 5, 5, Yes], [10, 12, 4, 4, Yes], [10, 13, 3, 3, Yes], [10, 14, 2, 2, Yes], [10, 15, 1, 1, Yes], [10, 16, 0, 0, Yes], [10, 17, 25, 25, Yes], [10, 18, 24, 24, Yes], [10, 19, 23, 23, Yes], [10, 20, 22, 22, Yes], [10, 21, 21, 21, Yes], [10, 22, 20, 20, Yes], [10, 23, 19, 19, Yes], [10, 24, 18, 18, Yes], [10, 25, 17, 17, Yes], [11, 0, 15, 15, Yes], [11, 1, 14, 14, Yes], [11, 2, 13, 13, Yes], [11, 3, 12, 12, Yes], [11, 4, 11, 11, Yes], [11, 5, 10, 10, Yes], [11, 6, 9, 9, Yes], [11, 7, 8, 8, Yes], [11, 8, 7, 7, Yes], [11, 9, 6, 6, Yes], [11, 10, 5, 5, Yes], [11, 11, 4, 4, Yes], [11, 12, 3, 3, Yes], [11, 13, 2, 2, Yes], [11, 14, 1, 1, Yes], [11, 15, 0, 0, Yes], [11, 16, 25, 25, Yes], [11, 17, 24, 24, Yes], [11, 18, 23, 23, Yes], [11, 19, 22, 22, Yes], [11, 20, 21, 21, Yes], [11, 21, 20, 20, Yes], [11, 22, 19, 19, Yes], [11, 23, 18, 18, Yes], [11, 24, 17, 17, Yes], [11, 25, 16, 16, Yes], [12, 0, 14, 14, Yes], [12, 1, 13, 13, Yes], [12, 2, 12, 12, Yes], [12, 3, 11, 11, Yes], [12, 4, 10, 10, Yes], [12, 5, 9, 9, Yes], [12, 6, 8, 8, Yes], [12, 7, 7, 7, Yes], [12, 8, 6, 6, Yes], [12, 9, 5, 5, Yes], [12, 10, 4, 4, Yes], [12, 11, 3, 3, Yes], [12, 12, 2, 2, Yes], [12, 13, 1, 1, Yes], [12, 14, 0, 0, Yes], [12, 15, 25, 25, Yes], [12, 16, 24, 24, Yes], [12, 17, 23, 23, Yes], [12, 18, 22, 22, Yes], [12, 19, 21, 21, Yes], [12, 20, 20, 20, Yes], [12, 21, 19, 19, Yes], [12, 22, 18, 18, Yes], [12, 23, 17, 17, Yes], [12, 24, 16, 16, Yes], [12, 25, 15, 15, Yes], [13, 0, 13, 13, Yes], [13, 1, 12, 12, Yes], [13, 2, 11, 11, Yes], [13, 3, 10, 10, Yes], [13, 4, 9, 9, Yes], [13, 5, 8, 8, Yes], [13, 6, 7, 7, Yes], [13, 7, 6, 6, Yes], [13, 8, 5, 5, Yes], [13, 9, 4, 4, Yes], [13, 10, 3, 3, Yes], [13, 11, 2, 2, Yes], [13, 12, 1, 1, Yes], [13, 13, 0, 0, Yes], [13, 14, 25, 25, Yes], [13, 15, 24, 24, Yes], [13, 16, 23, 23, Yes], [13, 17, 22, 22, Yes], [13, 18, 21, 21, Yes], [13, 19, 20, 20, Yes], [13, 20, 19, 19, Yes], [13, 21, 18, 18, Yes], [13, 22, 17, 17, Yes], [13, 23, 16, 16, Yes], [13, 24, 15, 15, Yes], [13, 25, 14, 14, Yes], [14, 0, 12, 12, Yes], [14, 1, 11, 11, Yes], [14, 2, 10, 10, Yes], [14, 3, 9, 9, Yes], [14, 4, 8, 8, Yes], [14, 5, 7, 7, Yes], [14, 6, 6, 6, Yes], [14, 7, 5, 5, Yes], [14, 8, 4, 4, Yes], [14, 9, 3, 3, Yes], [14, 10, 2, 2, Yes], [14, 11, 1, 1, Yes], [14, 12, 0, 0, Yes], [14, 13, 25, 25, Yes], [14, 14, 24, 24, Yes], [14, 15, 23, 23, Yes], [14, 16, 22, 22, Yes], [14, 17, 21, 21, Yes], [14, 18, 20, 20, Yes], [14, 19, 19, 19, Yes], [14, 20, 18, 18, Yes], [14, 21, 17, 17, Yes], [14, 22, 16, 16, Yes], [14, 23, 15, 15, Yes], [14, 24, 14, 14, Yes], [14, 25, 13, 13, Yes], [15, 0, 11, 11, Yes], [15, 1, 10, 10, Yes], [15, 2, 9, 9, Yes], [15, 3, 8, 8, Yes], [15, 4, 7, 7, Yes], [15, 5, 6, 6, Yes], [15, 6, 5, 5, Yes], [15, 7, 4, 4, Yes], [15, 8, 3, 3, Yes], [15, 9, 2, 2, Yes], [15, 10, 1, 1, Yes], [15, 11, 0, 0, Yes], [15, 12, 25, 25, Yes], [15, 13, 24, 24, Yes], [15, 14, 23, 23, Yes], [15, 15, 22, 22, Yes], [15, 16, 21, 21, Yes], [15, 17, 20, 20, Yes], [15, 18, 19, 19, Yes], [15, 19, 18, 18, Yes], [15, 20, 17, 17, Yes], [15, 21, 16, 16, Yes], [15, 22, 15, 15, Yes], [15, 23, 14, 14, Yes], [15, 24, 13, 13, Yes], [15, 25, 12, 12, Yes], [16, 0, 10, 10, Yes], [16, 1, 9, 9, Yes], [16, 2, 8, 8, Yes], [16, 3, 7, 7, Yes], [16, 4, 6, 6, Yes], [16, 5, 5, 5, Yes], [16, 6, 4, 4, Yes], [16, 7, 3, 3, Yes], [16, 8, 2, 2, Yes], [16, 9, 1, 1, Yes], [16, 10, 0, 0, Yes], [16, 11, 25, 25, Yes], [16, 12, 24, 24, Yes], [16, 13, 23, 23, Yes], [16, 14, 22, 22, Yes], [16, 15, 21, 21, Yes], [16, 16, 20, 20, Yes], [16, 17, 19, 19, Yes], [16, 18, 18, 18, Yes], [16, 19, 17, 17, Yes], [16, 20, 16, 16, Yes], [16, 21, 15, 15, Yes], [16, 22, 14, 14, Yes], [16, 23, 13, 13, Yes], [16, 24, 12, 12, Yes], [16, 25, 11, 11, Yes], [17, 0, 9, 9, Yes], [17, 1, 8, 8, Yes], [17, 2, 7, 7, Yes], [17, 3, 6, 6, Yes], [17, 4, 5, 5, Yes], [17, 5, 4, 4, Yes], [17, 6, 3, 3, Yes], [17, 7, 2, 2, Yes], [17, 8, 1, 1, Yes], [17, 9, 0, 0, Yes], [17, 10, 25, 25, Yes], [17, 11, 24, 24, Yes], [17, 12, 23, 23, Yes], [17, 13, 22, 22, Yes], [17, 14, 21, 21, Yes], [17, 15, 20, 20, Yes], [17, 16, 19, 19, Yes], [17, 17, 18, 18, Yes], [17, 18, 17, 17, Yes], [17, 19, 16, 16, Yes], [17, 20, 15, 15, Yes], [17, 21, 14, 14, Yes], [17, 22, 13, 13, Yes], [17, 23, 12, 12, Yes], [17, 24, 11, 11, Yes], [17, 25, 10, 10, Yes], [18, 0, 8, 8, Yes], [18, 1, 7, 7, Yes], [18, 2, 6, 6, Yes], [18, 3, 5, 5, Yes], [18, 4, 4, 4, Yes], [18, 5, 3, 3, Yes], [18, 6, 2, 2, Yes], [18, 7, 1, 1, Yes], [18, 8, 0, 0, Yes], [18, 9, 25, 25, Yes], [18, 10, 24, 24, Yes], [18, 11, 23, 23, Yes], [18, 12, 22, 22, Yes], [18, 13, 21, 21, Yes], [18, 14, 20, 20, Yes], [18, 15, 19, 19, Yes], [18, 16, 18, 18, Yes], [18, 17, 17, 17, Yes], [18, 18, 16, 16, Yes], [18, 19, 15, 15, Yes], [18, 20, 14, 14, Yes], [18, 21, 13, 13, Yes], [18, 22, 12, 12, Yes], [18, 23, 11, 11, Yes], [18, 24, 10, 10, Yes], [18, 25, 9, 9, Yes], [19, 0, 7, 7, Yes], [19, 1, 6, 6, Yes], [19, 2, 5, 5, Yes], [19, 3, 4, 4, Yes], [19, 4, 3, 3, Yes], [19, 5, 2, 2, Yes], [19, 6, 1, 1, Yes], [19, 7, 0, 0, Yes], [19, 8, 25, 25, Yes], [19, 9, 24, 24, Yes], [19, 10, 23, 23, Yes], [19, 11, 22, 22, Yes], [19, 12, 21, 21, Yes], [19, 13, 20, 20, Yes], [19, 14, 19, 19, Yes], [19, 15, 18, 18, Yes], [19, 16, 17, 17, Yes], [19, 17, 16, 16, Yes], [19, 18, 15, 15, Yes], [19, 19, 14, 14, Yes], [19, 20, 13, 13, Yes], [19, 21, 12, 12, Yes], [19, 22, 11, 11, Yes], [19, 23, 10, 10, Yes], [19, 24, 9, 9, Yes], [19, 25, 8, 8, Yes], [20, 0, 6, 6, Yes], [20, 1, 5, 5, Yes], [20, 2, 4, 4, Yes], [20, 3, 3, 3, Yes], [20, 4, 2, 2, Yes], [20, 5, 1, 1, Yes], [20, 6, 0, 0, Yes], [20, 7, 25, 25, Yes], [20, 8, 24, 24, Yes], [20, 9, 23, 23, Yes], [20, 10, 22, 22, Yes], [20, 11, 21, 21, Yes], [20, 12, 20, 20, Yes], [20, 13, 19, 19, Yes], [20, 14, 18, 18, Yes], [20, 15, 17, 17, Yes], [20, 16, 16, 16, Yes], [20, 17, 15, 15, Yes], [20, 18, 14, 14, Yes], [20, 19, 13, 13, Yes], [20, 20, 12, 12, Yes], [20, 21, 11, 11, Yes], [20, 22, 10, 10, Yes], [20, 23, 9, 9, Yes], [20, 24, 8, 8, Yes], [20, 25, 7, 7, Yes], [21, 0, 5, 5, Yes], [21, 1, 4, 4, Yes], [21, 2, 3, 3, Yes], [21, 3, 2, 2, Yes], [21, 4, 1, 1, Yes], [21, 5, 0, 0, Yes], [21, 6, 25, 25, Yes], [21, 7, 24, 24, Yes], [21, 8, 23, 23, Yes], [21, 9, 22, 22, Yes], [21, 10, 21, 21, Yes], [21, 11, 20, 20, Yes], [21, 12, 19, 19, Yes], [21, 13, 18, 18, Yes], [21, 14, 17, 17, Yes], [21, 15, 16, 16, Yes], [21, 16, 15, 15, Yes], [21, 17, 14, 14, Yes], [21, 18, 13, 13, Yes], [21, 19, 12, 12, Yes], [21, 20, 11, 11, Yes], [21, 21, 10, 10, Yes], [21, 22, 9, 9, Yes], [21, 23, 8, 8, Yes], [21, 24, 7, 7, Yes], [21, 25, 6, 6, Yes], [22, 0, 4, 4, Yes], [22, 1, 3, 3, Yes], [22, 2, 2, 2, Yes], [22, 3, 1, 1, Yes], [22, 4, 0, 0, Yes], [22, 5, 25, 25, Yes], [22, 6, 24, 24, Yes], [22, 7, 23, 23, Yes], [22, 8, 22, 22, Yes], [22, 9, 21, 21, Yes], [22, 10, 20, 20, Yes], [22, 11, 19, 19, Yes], [22, 12, 18, 18, Yes], [22, 13, 17, 17, Yes], [22, 14, 16, 16, Yes], [22, 15, 15, 15, Yes], [22, 16, 14, 14, Yes], [22, 17, 13, 13, Yes], [22, 18, 12, 12, Yes], [22, 19, 11, 11, Yes], [22, 20, 10, 10, Yes], [22, 21, 9, 9, Yes], [22, 22, 8, 8, Yes], [22, 23, 7, 7, Yes], [22, 24, 6, 6, Yes], [22, 25, 5, 5, Yes], [23, 0, 3, 3, Yes], [23, 1, 2, 2, Yes], [23, 2, 1, 1, Yes], [23, 3, 0, 0, Yes], [23, 4, 25, 25, Yes], [23, 5, 24, 24, Yes], [23, 6, 23, 23, Yes], [23, 7, 22, 22, Yes], [23, 8, 21, 21, Yes], [23, 9, 20, 20, Yes], [23, 10, 19, 19, Yes], [23, 11, 18, 18, Yes], [23, 12, 17, 17, Yes], [23, 13, 16, 16, Yes], [23, 14, 15, 15, Yes], [23, 15, 14, 14, Yes], [23, 16, 13, 13, Yes], [23, 17, 12, 12, Yes], [23, 18, 11, 11, Yes], [23, 19, 10, 10, Yes], [23, 20, 9, 9, Yes], [23, 21, 8, 8, Yes], [23, 22, 7, 7, Yes], [23, 23, 6, 6, Yes], [23, 24, 5, 5, Yes], [23, 25, 4, 4, Yes], [24, 0, 2, 2, Yes], [24, 1, 1, 1, Yes], [24, 2, 0, 0, Yes], [24, 3, 25, 25, Yes], [24, 4, 24, 24, Yes], [24, 5, 23, 23, Yes], [24, 6, 22, 22, Yes], [24, 7, 21, 21, Yes], [24, 8, 20, 20, Yes], [24, 9, 19, 19, Yes], [24, 10, 18, 18, Yes], [24, 11, 17, 17, Yes], [24, 12, 16, 16, Yes], [24, 13, 15, 15, Yes], [24, 14, 14, 14, Yes], [24, 15, 13, 13, Yes], [24, 16, 12, 12, Yes], [24, 17, 11, 11, Yes], [24, 18, 10, 10, Yes], [24, 19, 9, 9, Yes], [24, 20, 8, 8, Yes], [24, 21, 7, 7, Yes], [24, 22, 6, 6, Yes], [24, 23, 5, 5, Yes], [24, 24, 4, 4, Yes], [24, 25, 3, 3, Yes], [25, 0, 1, 1, Yes], [25, 1, 0, 0, Yes], [25, 2, 25, 25, Yes], [25, 3, 24, 24, Yes], [25, 4, 23, 23, Yes], [25, 5, 22, 22, Yes], [25, 6, 21, 21, Yes], [25, 7, 20, 20, Yes], [25, 8, 19, 19, Yes], [25, 9, 18, 18, Yes], [25, 10, 17, 17, Yes], [25, 11, 16, 16, Yes], [25, 12, 15, 15, Yes], [25, 13, 14, 14, Yes], [25, 14, 13, 13, Yes], [25, 15, 12, 12, Yes], [25, 16, 11, 11, Yes], [25, 17, 10, 10, Yes], [25, 18, 9, 9, Yes], [25, 19, 8, 8, Yes], [25, 20, 7, 7, Yes], [25, 21, 6, 6, Yes], [25, 22, 5, 5, Yes], [25, 23, 4, 4, Yes], [25, 24, 3, 3, Yes], [25, 25, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-homomorphism-qualification · candidate
25. Qualify Both Groups and Every Product Pair before Applying Kernel Theorems
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]], target identity=0, and total mapping=[0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate).
View answer and explanation
(27, 27, [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 26, 26, Yes], [0, 2, 25, 25, Yes], [0, 3, 24, 24, Yes], [0, 4, 23, 23, Yes], [0, 5, 22, 22, Yes], [0, 6, 21, 21, Yes], [0, 7, 20, 20, Yes], [0, 8, 19, 19, Yes], [0, 9, 18, 18, Yes], [0, 10, 17, 17, Yes], [0, 11, 16, 16, Yes], [0, 12, 15, 15, Yes], [0, 13, 14, 14, Yes], [0, 14, 13, 13, Yes], [0, 15, 12, 12, Yes], [0, 16, 11, 11, Yes], [0, 17, 10, 10, Yes], [0, 18, 9, 9, Yes], [0, 19, 8, 8, Yes], [0, 20, 7, 7, Yes], [0, 21, 6, 6, Yes], [0, 22, 5, 5, Yes], [0, 23, 4, 4, Yes], [0, 24, 3, 3, Yes], [0, 25, 2, 2, Yes], [0, 26, 1, 1, Yes], [1, 0, 26, 26, Yes], [1, 1, 25, 25, Yes], [1, 2, 24, 24, Yes], [1, 3, 23, 23, Yes], [1, 4, 22, 22, Yes], [1, 5, 21, 21, Yes], [1, 6, 20, 20, Yes], [1, 7, 19, 19, Yes], [1, 8, 18, 18, Yes], [1, 9, 17, 17, Yes], [1, 10, 16, 16, Yes], [1, 11, 15, 15, Yes], [1, 12, 14, 14, Yes], [1, 13, 13, 13, Yes], [1, 14, 12, 12, Yes], [1, 15, 11, 11, Yes], [1, 16, 10, 10, Yes], [1, 17, 9, 9, Yes], [1, 18, 8, 8, Yes], [1, 19, 7, 7, Yes], [1, 20, 6, 6, Yes], [1, 21, 5, 5, Yes], [1, 22, 4, 4, Yes], [1, 23, 3, 3, Yes], [1, 24, 2, 2, Yes], [1, 25, 1, 1, Yes], [1, 26, 0, 0, Yes], [2, 0, 25, 25, Yes], [2, 1, 24, 24, Yes], [2, 2, 23, 23, Yes], [2, 3, 22, 22, Yes], [2, 4, 21, 21, Yes], [2, 5, 20, 20, Yes], [2, 6, 19, 19, Yes], [2, 7, 18, 18, Yes], [2, 8, 17, 17, Yes], [2, 9, 16, 16, Yes], [2, 10, 15, 15, Yes], [2, 11, 14, 14, Yes], [2, 12, 13, 13, Yes], [2, 13, 12, 12, Yes], [2, 14, 11, 11, Yes], [2, 15, 10, 10, Yes], [2, 16, 9, 9, Yes], [2, 17, 8, 8, Yes], [2, 18, 7, 7, Yes], [2, 19, 6, 6, Yes], [2, 20, 5, 5, Yes], [2, 21, 4, 4, Yes], [2, 22, 3, 3, Yes], [2, 23, 2, 2, Yes], [2, 24, 1, 1, Yes], [2, 25, 0, 0, Yes], [2, 26, 26, 26, Yes], [3, 0, 24, 24, Yes], [3, 1, 23, 23, Yes], [3, 2, 22, 22, Yes], [3, 3, 21, 21, Yes], [3, 4, 20, 20, Yes], [3, 5, 19, 19, Yes], [3, 6, 18, 18, Yes], [3, 7, 17, 17, Yes], [3, 8, 16, 16, Yes], [3, 9, 15, 15, Yes], [3, 10, 14, 14, Yes], [3, 11, 13, 13, Yes], [3, 12, 12, 12, Yes], [3, 13, 11, 11, Yes], [3, 14, 10, 10, Yes], [3, 15, 9, 9, Yes], [3, 16, 8, 8, Yes], [3, 17, 7, 7, Yes], [3, 18, 6, 6, Yes], [3, 19, 5, 5, Yes], [3, 20, 4, 4, Yes], [3, 21, 3, 3, Yes], [3, 22, 2, 2, Yes], [3, 23, 1, 1, Yes], [3, 24, 0, 0, Yes], [3, 25, 26, 26, Yes], [3, 26, 25, 25, Yes], [4, 0, 23, 23, Yes], [4, 1, 22, 22, Yes], [4, 2, 21, 21, Yes], [4, 3, 20, 20, Yes], [4, 4, 19, 19, Yes], [4, 5, 18, 18, Yes], [4, 6, 17, 17, Yes], [4, 7, 16, 16, Yes], [4, 8, 15, 15, Yes], [4, 9, 14, 14, Yes], [4, 10, 13, 13, Yes], [4, 11, 12, 12, Yes], [4, 12, 11, 11, Yes], [4, 13, 10, 10, Yes], [4, 14, 9, 9, Yes], [4, 15, 8, 8, Yes], [4, 16, 7, 7, Yes], [4, 17, 6, 6, Yes], [4, 18, 5, 5, Yes], [4, 19, 4, 4, Yes], [4, 20, 3, 3, Yes], [4, 21, 2, 2, Yes], [4, 22, 1, 1, Yes], [4, 23, 0, 0, Yes], [4, 24, 26, 26, Yes], [4, 25, 25, 25, Yes], [4, 26, 24, 24, Yes], [5, 0, 22, 22, Yes], [5, 1, 21, 21, Yes], [5, 2, 20, 20, Yes], [5, 3, 19, 19, Yes], [5, 4, 18, 18, Yes], [5, 5, 17, 17, Yes], [5, 6, 16, 16, Yes], [5, 7, 15, 15, Yes], [5, 8, 14, 14, Yes], [5, 9, 13, 13, Yes], [5, 10, 12, 12, Yes], [5, 11, 11, 11, Yes], [5, 12, 10, 10, Yes], [5, 13, 9, 9, Yes], [5, 14, 8, 8, Yes], [5, 15, 7, 7, Yes], [5, 16, 6, 6, Yes], [5, 17, 5, 5, Yes], [5, 18, 4, 4, Yes], [5, 19, 3, 3, Yes], [5, 20, 2, 2, Yes], [5, 21, 1, 1, Yes], [5, 22, 0, 0, Yes], [5, 23, 26, 26, Yes], [5, 24, 25, 25, Yes], [5, 25, 24, 24, Yes], [5, 26, 23, 23, Yes], [6, 0, 21, 21, Yes], [6, 1, 20, 20, Yes], [6, 2, 19, 19, Yes], [6, 3, 18, 18, Yes], [6, 4, 17, 17, Yes], [6, 5, 16, 16, Yes], [6, 6, 15, 15, Yes], [6, 7, 14, 14, Yes], [6, 8, 13, 13, Yes], [6, 9, 12, 12, Yes], [6, 10, 11, 11, Yes], [6, 11, 10, 10, Yes], [6, 12, 9, 9, Yes], [6, 13, 8, 8, Yes], [6, 14, 7, 7, Yes], [6, 15, 6, 6, Yes], [6, 16, 5, 5, Yes], [6, 17, 4, 4, Yes], [6, 18, 3, 3, Yes], [6, 19, 2, 2, Yes], [6, 20, 1, 1, Yes], [6, 21, 0, 0, Yes], [6, 22, 26, 26, Yes], [6, 23, 25, 25, Yes], [6, 24, 24, 24, Yes], [6, 25, 23, 23, Yes], [6, 26, 22, 22, Yes], [7, 0, 20, 20, Yes], [7, 1, 19, 19, Yes], [7, 2, 18, 18, Yes], [7, 3, 17, 17, Yes], [7, 4, 16, 16, Yes], [7, 5, 15, 15, Yes], [7, 6, 14, 14, Yes], [7, 7, 13, 13, Yes], [7, 8, 12, 12, Yes], [7, 9, 11, 11, Yes], [7, 10, 10, 10, Yes], [7, 11, 9, 9, Yes], [7, 12, 8, 8, Yes], [7, 13, 7, 7, Yes], [7, 14, 6, 6, Yes], [7, 15, 5, 5, Yes], [7, 16, 4, 4, Yes], [7, 17, 3, 3, Yes], [7, 18, 2, 2, Yes], [7, 19, 1, 1, Yes], [7, 20, 0, 0, Yes], [7, 21, 26, 26, Yes], [7, 22, 25, 25, Yes], [7, 23, 24, 24, Yes], [7, 24, 23, 23, Yes], [7, 25, 22, 22, Yes], [7, 26, 21, 21, Yes], [8, 0, 19, 19, Yes], [8, 1, 18, 18, Yes], [8, 2, 17, 17, Yes], [8, 3, 16, 16, Yes], [8, 4, 15, 15, Yes], [8, 5, 14, 14, Yes], [8, 6, 13, 13, Yes], [8, 7, 12, 12, Yes], [8, 8, 11, 11, Yes], [8, 9, 10, 10, Yes], [8, 10, 9, 9, Yes], [8, 11, 8, 8, Yes], [8, 12, 7, 7, Yes], [8, 13, 6, 6, Yes], [8, 14, 5, 5, Yes], [8, 15, 4, 4, Yes], [8, 16, 3, 3, Yes], [8, 17, 2, 2, Yes], [8, 18, 1, 1, Yes], [8, 19, 0, 0, Yes], [8, 20, 26, 26, Yes], [8, 21, 25, 25, Yes], [8, 22, 24, 24, Yes], [8, 23, 23, 23, Yes], [8, 24, 22, 22, Yes], [8, 25, 21, 21, Yes], [8, 26, 20, 20, Yes], [9, 0, 18, 18, Yes], [9, 1, 17, 17, Yes], [9, 2, 16, 16, Yes], [9, 3, 15, 15, Yes], [9, 4, 14, 14, Yes], [9, 5, 13, 13, Yes], [9, 6, 12, 12, Yes], [9, 7, 11, 11, Yes], [9, 8, 10, 10, Yes], [9, 9, 9, 9, Yes], [9, 10, 8, 8, Yes], [9, 11, 7, 7, Yes], [9, 12, 6, 6, Yes], [9, 13, 5, 5, Yes], [9, 14, 4, 4, Yes], [9, 15, 3, 3, Yes], [9, 16, 2, 2, Yes], [9, 17, 1, 1, Yes], [9, 18, 0, 0, Yes], [9, 19, 26, 26, Yes], [9, 20, 25, 25, Yes], [9, 21, 24, 24, Yes], [9, 22, 23, 23, Yes], [9, 23, 22, 22, Yes], [9, 24, 21, 21, Yes], [9, 25, 20, 20, Yes], [9, 26, 19, 19, Yes], [10, 0, 17, 17, Yes], [10, 1, 16, 16, Yes], [10, 2, 15, 15, Yes], [10, 3, 14, 14, Yes], [10, 4, 13, 13, Yes], [10, 5, 12, 12, Yes], [10, 6, 11, 11, Yes], [10, 7, 10, 10, Yes], [10, 8, 9, 9, Yes], [10, 9, 8, 8, Yes], [10, 10, 7, 7, Yes], [10, 11, 6, 6, Yes], [10, 12, 5, 5, Yes], [10, 13, 4, 4, Yes], [10, 14, 3, 3, Yes], [10, 15, 2, 2, Yes], [10, 16, 1, 1, Yes], [10, 17, 0, 0, Yes], [10, 18, 26, 26, Yes], [10, 19, 25, 25, Yes], [10, 20, 24, 24, Yes], [10, 21, 23, 23, Yes], [10, 22, 22, 22, Yes], [10, 23, 21, 21, Yes], [10, 24, 20, 20, Yes], [10, 25, 19, 19, Yes], [10, 26, 18, 18, Yes], [11, 0, 16, 16, Yes], [11, 1, 15, 15, Yes], [11, 2, 14, 14, Yes], [11, 3, 13, 13, Yes], [11, 4, 12, 12, Yes], [11, 5, 11, 11, Yes], [11, 6, 10, 10, Yes], [11, 7, 9, 9, Yes], [11, 8, 8, 8, Yes], [11, 9, 7, 7, Yes], [11, 10, 6, 6, Yes], [11, 11, 5, 5, Yes], [11, 12, 4, 4, Yes], [11, 13, 3, 3, Yes], [11, 14, 2, 2, Yes], [11, 15, 1, 1, Yes], [11, 16, 0, 0, Yes], [11, 17, 26, 26, Yes], [11, 18, 25, 25, Yes], [11, 19, 24, 24, Yes], [11, 20, 23, 23, Yes], [11, 21, 22, 22, Yes], [11, 22, 21, 21, Yes], [11, 23, 20, 20, Yes], [11, 24, 19, 19, Yes], [11, 25, 18, 18, Yes], [11, 26, 17, 17, Yes], [12, 0, 15, 15, Yes], [12, 1, 14, 14, Yes], [12, 2, 13, 13, Yes], [12, 3, 12, 12, Yes], [12, 4, 11, 11, Yes], [12, 5, 10, 10, Yes], [12, 6, 9, 9, Yes], [12, 7, 8, 8, Yes], [12, 8, 7, 7, Yes], [12, 9, 6, 6, Yes], [12, 10, 5, 5, Yes], [12, 11, 4, 4, Yes], [12, 12, 3, 3, Yes], [12, 13, 2, 2, Yes], [12, 14, 1, 1, Yes], [12, 15, 0, 0, Yes], [12, 16, 26, 26, Yes], [12, 17, 25, 25, Yes], [12, 18, 24, 24, Yes], [12, 19, 23, 23, Yes], [12, 20, 22, 22, Yes], [12, 21, 21, 21, Yes], [12, 22, 20, 20, Yes], [12, 23, 19, 19, Yes], [12, 24, 18, 18, Yes], [12, 25, 17, 17, Yes], [12, 26, 16, 16, Yes], [13, 0, 14, 14, Yes], [13, 1, 13, 13, Yes], [13, 2, 12, 12, Yes], [13, 3, 11, 11, Yes], [13, 4, 10, 10, Yes], [13, 5, 9, 9, Yes], [13, 6, 8, 8, Yes], [13, 7, 7, 7, Yes], [13, 8, 6, 6, Yes], [13, 9, 5, 5, Yes], [13, 10, 4, 4, Yes], [13, 11, 3, 3, Yes], [13, 12, 2, 2, Yes], [13, 13, 1, 1, Yes], [13, 14, 0, 0, Yes], [13, 15, 26, 26, Yes], [13, 16, 25, 25, Yes], [13, 17, 24, 24, Yes], [13, 18, 23, 23, Yes], [13, 19, 22, 22, Yes], [13, 20, 21, 21, Yes], [13, 21, 20, 20, Yes], [13, 22, 19, 19, Yes], [13, 23, 18, 18, Yes], [13, 24, 17, 17, Yes], [13, 25, 16, 16, Yes], [13, 26, 15, 15, Yes], [14, 0, 13, 13, Yes], [14, 1, 12, 12, Yes], [14, 2, 11, 11, Yes], [14, 3, 10, 10, Yes], [14, 4, 9, 9, Yes], [14, 5, 8, 8, Yes], [14, 6, 7, 7, Yes], [14, 7, 6, 6, Yes], [14, 8, 5, 5, Yes], [14, 9, 4, 4, Yes], [14, 10, 3, 3, Yes], [14, 11, 2, 2, Yes], [14, 12, 1, 1, Yes], [14, 13, 0, 0, Yes], [14, 14, 26, 26, Yes], [14, 15, 25, 25, Yes], [14, 16, 24, 24, Yes], [14, 17, 23, 23, Yes], [14, 18, 22, 22, Yes], [14, 19, 21, 21, Yes], [14, 20, 20, 20, Yes], [14, 21, 19, 19, Yes], [14, 22, 18, 18, Yes], [14, 23, 17, 17, Yes], [14, 24, 16, 16, Yes], [14, 25, 15, 15, Yes], [14, 26, 14, 14, Yes], [15, 0, 12, 12, Yes], [15, 1, 11, 11, Yes], [15, 2, 10, 10, Yes], [15, 3, 9, 9, Yes], [15, 4, 8, 8, Yes], [15, 5, 7, 7, Yes], [15, 6, 6, 6, Yes], [15, 7, 5, 5, Yes], [15, 8, 4, 4, Yes], [15, 9, 3, 3, Yes], [15, 10, 2, 2, Yes], [15, 11, 1, 1, Yes], [15, 12, 0, 0, Yes], [15, 13, 26, 26, Yes], [15, 14, 25, 25, Yes], [15, 15, 24, 24, Yes], [15, 16, 23, 23, Yes], [15, 17, 22, 22, Yes], [15, 18, 21, 21, Yes], [15, 19, 20, 20, Yes], [15, 20, 19, 19, Yes], [15, 21, 18, 18, Yes], [15, 22, 17, 17, Yes], [15, 23, 16, 16, Yes], [15, 24, 15, 15, Yes], [15, 25, 14, 14, Yes], [15, 26, 13, 13, Yes], [16, 0, 11, 11, Yes], [16, 1, 10, 10, Yes], [16, 2, 9, 9, Yes], [16, 3, 8, 8, Yes], [16, 4, 7, 7, Yes], [16, 5, 6, 6, Yes], [16, 6, 5, 5, Yes], [16, 7, 4, 4, Yes], [16, 8, 3, 3, Yes], [16, 9, 2, 2, Yes], [16, 10, 1, 1, Yes], [16, 11, 0, 0, Yes], [16, 12, 26, 26, Yes], [16, 13, 25, 25, Yes], [16, 14, 24, 24, Yes], [16, 15, 23, 23, Yes], [16, 16, 22, 22, Yes], [16, 17, 21, 21, Yes], [16, 18, 20, 20, Yes], [16, 19, 19, 19, Yes], [16, 20, 18, 18, Yes], [16, 21, 17, 17, Yes], [16, 22, 16, 16, Yes], [16, 23, 15, 15, Yes], [16, 24, 14, 14, Yes], [16, 25, 13, 13, Yes], [16, 26, 12, 12, Yes], [17, 0, 10, 10, Yes], [17, 1, 9, 9, Yes], [17, 2, 8, 8, Yes], [17, 3, 7, 7, Yes], [17, 4, 6, 6, Yes], [17, 5, 5, 5, Yes], [17, 6, 4, 4, Yes], [17, 7, 3, 3, Yes], [17, 8, 2, 2, Yes], [17, 9, 1, 1, Yes], [17, 10, 0, 0, Yes], [17, 11, 26, 26, Yes], [17, 12, 25, 25, Yes], [17, 13, 24, 24, Yes], [17, 14, 23, 23, Yes], [17, 15, 22, 22, Yes], [17, 16, 21, 21, Yes], [17, 17, 20, 20, Yes], [17, 18, 19, 19, Yes], [17, 19, 18, 18, Yes], [17, 20, 17, 17, Yes], [17, 21, 16, 16, Yes], [17, 22, 15, 15, Yes], [17, 23, 14, 14, Yes], [17, 24, 13, 13, Yes], [17, 25, 12, 12, Yes], [17, 26, 11, 11, Yes], [18, 0, 9, 9, Yes], [18, 1, 8, 8, Yes], [18, 2, 7, 7, Yes], [18, 3, 6, 6, Yes], [18, 4, 5, 5, Yes], [18, 5, 4, 4, Yes], [18, 6, 3, 3, Yes], [18, 7, 2, 2, Yes], [18, 8, 1, 1, Yes], [18, 9, 0, 0, Yes], [18, 10, 26, 26, Yes], [18, 11, 25, 25, Yes], [18, 12, 24, 24, Yes], [18, 13, 23, 23, Yes], [18, 14, 22, 22, Yes], [18, 15, 21, 21, Yes], [18, 16, 20, 20, Yes], [18, 17, 19, 19, Yes], [18, 18, 18, 18, Yes], [18, 19, 17, 17, Yes], [18, 20, 16, 16, Yes], [18, 21, 15, 15, Yes], [18, 22, 14, 14, Yes], [18, 23, 13, 13, Yes], [18, 24, 12, 12, Yes], [18, 25, 11, 11, Yes], [18, 26, 10, 10, Yes], [19, 0, 8, 8, Yes], [19, 1, 7, 7, Yes], [19, 2, 6, 6, Yes], [19, 3, 5, 5, Yes], [19, 4, 4, 4, Yes], [19, 5, 3, 3, Yes], [19, 6, 2, 2, Yes], [19, 7, 1, 1, Yes], [19, 8, 0, 0, Yes], [19, 9, 26, 26, Yes], [19, 10, 25, 25, Yes], [19, 11, 24, 24, Yes], [19, 12, 23, 23, Yes], [19, 13, 22, 22, Yes], [19, 14, 21, 21, Yes], [19, 15, 20, 20, Yes], [19, 16, 19, 19, Yes], [19, 17, 18, 18, Yes], [19, 18, 17, 17, Yes], [19, 19, 16, 16, Yes], [19, 20, 15, 15, Yes], [19, 21, 14, 14, Yes], [19, 22, 13, 13, Yes], [19, 23, 12, 12, Yes], [19, 24, 11, 11, Yes], [19, 25, 10, 10, Yes], [19, 26, 9, 9, Yes], [20, 0, 7, 7, Yes], [20, 1, 6, 6, Yes], [20, 2, 5, 5, Yes], [20, 3, 4, 4, Yes], [20, 4, 3, 3, Yes], [20, 5, 2, 2, Yes], [20, 6, 1, 1, Yes], [20, 7, 0, 0, Yes], [20, 8, 26, 26, Yes], [20, 9, 25, 25, Yes], [20, 10, 24, 24, Yes], [20, 11, 23, 23, Yes], [20, 12, 22, 22, Yes], [20, 13, 21, 21, Yes], [20, 14, 20, 20, Yes], [20, 15, 19, 19, Yes], [20, 16, 18, 18, Yes], [20, 17, 17, 17, Yes], [20, 18, 16, 16, Yes], [20, 19, 15, 15, Yes], [20, 20, 14, 14, Yes], [20, 21, 13, 13, Yes], [20, 22, 12, 12, Yes], [20, 23, 11, 11, Yes], [20, 24, 10, 10, Yes], [20, 25, 9, 9, Yes], [20, 26, 8, 8, Yes], [21, 0, 6, 6, Yes], [21, 1, 5, 5, Yes], [21, 2, 4, 4, Yes], [21, 3, 3, 3, Yes], [21, 4, 2, 2, Yes], [21, 5, 1, 1, Yes], [21, 6, 0, 0, Yes], [21, 7, 26, 26, Yes], [21, 8, 25, 25, Yes], [21, 9, 24, 24, Yes], [21, 10, 23, 23, Yes], [21, 11, 22, 22, Yes], [21, 12, 21, 21, Yes], [21, 13, 20, 20, Yes], [21, 14, 19, 19, Yes], [21, 15, 18, 18, Yes], [21, 16, 17, 17, Yes], [21, 17, 16, 16, Yes], [21, 18, 15, 15, Yes], [21, 19, 14, 14, Yes], [21, 20, 13, 13, Yes], [21, 21, 12, 12, Yes], [21, 22, 11, 11, Yes], [21, 23, 10, 10, Yes], [21, 24, 9, 9, Yes], [21, 25, 8, 8, Yes], [21, 26, 7, 7, Yes], [22, 0, 5, 5, Yes], [22, 1, 4, 4, Yes], [22, 2, 3, 3, Yes], [22, 3, 2, 2, Yes], [22, 4, 1, 1, Yes], [22, 5, 0, 0, Yes], [22, 6, 26, 26, Yes], [22, 7, 25, 25, Yes], [22, 8, 24, 24, Yes], [22, 9, 23, 23, Yes], [22, 10, 22, 22, Yes], [22, 11, 21, 21, Yes], [22, 12, 20, 20, Yes], [22, 13, 19, 19, Yes], [22, 14, 18, 18, Yes], [22, 15, 17, 17, Yes], [22, 16, 16, 16, Yes], [22, 17, 15, 15, Yes], [22, 18, 14, 14, Yes], [22, 19, 13, 13, Yes], [22, 20, 12, 12, Yes], [22, 21, 11, 11, Yes], [22, 22, 10, 10, Yes], [22, 23, 9, 9, Yes], [22, 24, 8, 8, Yes], [22, 25, 7, 7, Yes], [22, 26, 6, 6, Yes], [23, 0, 4, 4, Yes], [23, 1, 3, 3, Yes], [23, 2, 2, 2, Yes], [23, 3, 1, 1, Yes], [23, 4, 0, 0, Yes], [23, 5, 26, 26, Yes], [23, 6, 25, 25, Yes], [23, 7, 24, 24, Yes], [23, 8, 23, 23, Yes], [23, 9, 22, 22, Yes], [23, 10, 21, 21, Yes], [23, 11, 20, 20, Yes], [23, 12, 19, 19, Yes], [23, 13, 18, 18, Yes], [23, 14, 17, 17, Yes], [23, 15, 16, 16, Yes], [23, 16, 15, 15, Yes], [23, 17, 14, 14, Yes], [23, 18, 13, 13, Yes], [23, 19, 12, 12, Yes], [23, 20, 11, 11, Yes], [23, 21, 10, 10, Yes], [23, 22, 9, 9, Yes], [23, 23, 8, 8, Yes], [23, 24, 7, 7, Yes], [23, 25, 6, 6, Yes], [23, 26, 5, 5, Yes], [24, 0, 3, 3, Yes], [24, 1, 2, 2, Yes], [24, 2, 1, 1, Yes], [24, 3, 0, 0, Yes], [24, 4, 26, 26, Yes], [24, 5, 25, 25, Yes], [24, 6, 24, 24, Yes], [24, 7, 23, 23, Yes], [24, 8, 22, 22, Yes], [24, 9, 21, 21, Yes], [24, 10, 20, 20, Yes], [24, 11, 19, 19, Yes], [24, 12, 18, 18, Yes], [24, 13, 17, 17, Yes], [24, 14, 16, 16, Yes], [24, 15, 15, 15, Yes], [24, 16, 14, 14, Yes], [24, 17, 13, 13, Yes], [24, 18, 12, 12, Yes], [24, 19, 11, 11, Yes], [24, 20, 10, 10, Yes], [24, 21, 9, 9, Yes], [24, 22, 8, 8, Yes], [24, 23, 7, 7, Yes], [24, 24, 6, 6, Yes], [24, 25, 5, 5, Yes], [24, 26, 4, 4, Yes], [25, 0, 2, 2, Yes], [25, 1, 1, 1, Yes], [25, 2, 0, 0, Yes], [25, 3, 26, 26, Yes], [25, 4, 25, 25, Yes], [25, 5, 24, 24, Yes], [25, 6, 23, 23, Yes], [25, 7, 22, 22, Yes], [25, 8, 21, 21, Yes], [25, 9, 20, 20, Yes], [25, 10, 19, 19, Yes], [25, 11, 18, 18, Yes], [25, 12, 17, 17, Yes], [25, 13, 16, 16, Yes], [25, 14, 15, 15, Yes], [25, 15, 14, 14, Yes], [25, 16, 13, 13, Yes], [25, 17, 12, 12, Yes], [25, 18, 11, 11, Yes], [25, 19, 10, 10, Yes], [25, 20, 9, 9, Yes], [25, 21, 8, 8, Yes], [25, 22, 7, 7, Yes], [25, 23, 6, 6, Yes], [25, 24, 5, 5, Yes], [25, 25, 4, 4, Yes], [25, 26, 3, 3, Yes], [26, 0, 1, 1, Yes], [26, 1, 0, 0, Yes], [26, 2, 26, 26, Yes], [26, 3, 25, 25, Yes], [26, 4, 24, 24, Yes], [26, 5, 23, 23, Yes], [26, 6, 22, 22, Yes], [26, 7, 21, 21, Yes], [26, 8, 20, 20, Yes], [26, 9, 19, 19, Yes], [26, 10, 18, 18, Yes], [26, 11, 17, 17, Yes], [26, 12, 16, 16, Yes], [26, 13, 15, 15, Yes], [26, 14, 14, 14, Yes], [26, 15, 13, 13, Yes], [26, 16, 12, 12, Yes], [26, 17, 11, 11, Yes], [26, 18, 10, 10, Yes], [26, 19, 9, 9, Yes], [26, 20, 8, 8, Yes], [26, 21, 7, 7, Yes], [26, 22, 6, 6, Yes], [26, 23, 5, 5, Yes], [26, 24, 4, 4, Yes], [26, 25, 3, 3, Yes], [26, 26, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair)
List [x,y,f(x*y),f(x)*f(y),equal] for every source pair. Report the complete homomorphism flag and first failed pair or null. A failed total map blocks normal-kernel, fiber and injectivity theorems; do not silently certify its identity preimage as a normal subgroup. The exact result, retaining the requested order, is (27, 27, [0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0, Yes], [0, 1, 26, 26, Yes], [0, 2, 25, 25, Yes], [0, 3, 24, 24, Yes], [0, 4, 23, 23, Yes], [0, 5, 22, 22, Yes], [0, 6, 21, 21, Yes], [0, 7, 20, 20, Yes], [0, 8, 19, 19, Yes], [0, 9, 18, 18, Yes], [0, 10, 17, 17, Yes], [0, 11, 16, 16, Yes], [0, 12, 15, 15, Yes], [0, 13, 14, 14, Yes], [0, 14, 13, 13, Yes], [0, 15, 12, 12, Yes], [0, 16, 11, 11, Yes], [0, 17, 10, 10, Yes], [0, 18, 9, 9, Yes], [0, 19, 8, 8, Yes], [0, 20, 7, 7, Yes], [0, 21, 6, 6, Yes], [0, 22, 5, 5, Yes], [0, 23, 4, 4, Yes], [0, 24, 3, 3, Yes], [0, 25, 2, 2, Yes], [0, 26, 1, 1, Yes], [1, 0, 26, 26, Yes], [1, 1, 25, 25, Yes], [1, 2, 24, 24, Yes], [1, 3, 23, 23, Yes], [1, 4, 22, 22, Yes], [1, 5, 21, 21, Yes], [1, 6, 20, 20, Yes], [1, 7, 19, 19, Yes], [1, 8, 18, 18, Yes], [1, 9, 17, 17, Yes], [1, 10, 16, 16, Yes], [1, 11, 15, 15, Yes], [1, 12, 14, 14, Yes], [1, 13, 13, 13, Yes], [1, 14, 12, 12, Yes], [1, 15, 11, 11, Yes], [1, 16, 10, 10, Yes], [1, 17, 9, 9, Yes], [1, 18, 8, 8, Yes], [1, 19, 7, 7, Yes], [1, 20, 6, 6, Yes], [1, 21, 5, 5, Yes], [1, 22, 4, 4, Yes], [1, 23, 3, 3, Yes], [1, 24, 2, 2, Yes], [1, 25, 1, 1, Yes], [1, 26, 0, 0, Yes], [2, 0, 25, 25, Yes], [2, 1, 24, 24, Yes], [2, 2, 23, 23, Yes], [2, 3, 22, 22, Yes], [2, 4, 21, 21, Yes], [2, 5, 20, 20, Yes], [2, 6, 19, 19, Yes], [2, 7, 18, 18, Yes], [2, 8, 17, 17, Yes], [2, 9, 16, 16, Yes], [2, 10, 15, 15, Yes], [2, 11, 14, 14, Yes], [2, 12, 13, 13, Yes], [2, 13, 12, 12, Yes], [2, 14, 11, 11, Yes], [2, 15, 10, 10, Yes], [2, 16, 9, 9, Yes], [2, 17, 8, 8, Yes], [2, 18, 7, 7, Yes], [2, 19, 6, 6, Yes], [2, 20, 5, 5, Yes], [2, 21, 4, 4, Yes], [2, 22, 3, 3, Yes], [2, 23, 2, 2, Yes], [2, 24, 1, 1, Yes], [2, 25, 0, 0, Yes], [2, 26, 26, 26, Yes], [3, 0, 24, 24, Yes], [3, 1, 23, 23, Yes], [3, 2, 22, 22, Yes], [3, 3, 21, 21, Yes], [3, 4, 20, 20, Yes], [3, 5, 19, 19, Yes], [3, 6, 18, 18, Yes], [3, 7, 17, 17, Yes], [3, 8, 16, 16, Yes], [3, 9, 15, 15, Yes], [3, 10, 14, 14, Yes], [3, 11, 13, 13, Yes], [3, 12, 12, 12, Yes], [3, 13, 11, 11, Yes], [3, 14, 10, 10, Yes], [3, 15, 9, 9, Yes], [3, 16, 8, 8, Yes], [3, 17, 7, 7, Yes], [3, 18, 6, 6, Yes], [3, 19, 5, 5, Yes], [3, 20, 4, 4, Yes], [3, 21, 3, 3, Yes], [3, 22, 2, 2, Yes], [3, 23, 1, 1, Yes], [3, 24, 0, 0, Yes], [3, 25, 26, 26, Yes], [3, 26, 25, 25, Yes], [4, 0, 23, 23, Yes], [4, 1, 22, 22, Yes], [4, 2, 21, 21, Yes], [4, 3, 20, 20, Yes], [4, 4, 19, 19, Yes], [4, 5, 18, 18, Yes], [4, 6, 17, 17, Yes], [4, 7, 16, 16, Yes], [4, 8, 15, 15, Yes], [4, 9, 14, 14, Yes], [4, 10, 13, 13, Yes], [4, 11, 12, 12, Yes], [4, 12, 11, 11, Yes], [4, 13, 10, 10, Yes], [4, 14, 9, 9, Yes], [4, 15, 8, 8, Yes], [4, 16, 7, 7, Yes], [4, 17, 6, 6, Yes], [4, 18, 5, 5, Yes], [4, 19, 4, 4, Yes], [4, 20, 3, 3, Yes], [4, 21, 2, 2, Yes], [4, 22, 1, 1, Yes], [4, 23, 0, 0, Yes], [4, 24, 26, 26, Yes], [4, 25, 25, 25, Yes], [4, 26, 24, 24, Yes], [5, 0, 22, 22, Yes], [5, 1, 21, 21, Yes], [5, 2, 20, 20, Yes], [5, 3, 19, 19, Yes], [5, 4, 18, 18, Yes], [5, 5, 17, 17, Yes], [5, 6, 16, 16, Yes], [5, 7, 15, 15, Yes], [5, 8, 14, 14, Yes], [5, 9, 13, 13, Yes], [5, 10, 12, 12, Yes], [5, 11, 11, 11, Yes], [5, 12, 10, 10, Yes], [5, 13, 9, 9, Yes], [5, 14, 8, 8, Yes], [5, 15, 7, 7, Yes], [5, 16, 6, 6, Yes], [5, 17, 5, 5, Yes], [5, 18, 4, 4, Yes], [5, 19, 3, 3, Yes], [5, 20, 2, 2, Yes], [5, 21, 1, 1, Yes], [5, 22, 0, 0, Yes], [5, 23, 26, 26, Yes], [5, 24, 25, 25, Yes], [5, 25, 24, 24, Yes], [5, 26, 23, 23, Yes], [6, 0, 21, 21, Yes], [6, 1, 20, 20, Yes], [6, 2, 19, 19, Yes], [6, 3, 18, 18, Yes], [6, 4, 17, 17, Yes], [6, 5, 16, 16, Yes], [6, 6, 15, 15, Yes], [6, 7, 14, 14, Yes], [6, 8, 13, 13, Yes], [6, 9, 12, 12, Yes], [6, 10, 11, 11, Yes], [6, 11, 10, 10, Yes], [6, 12, 9, 9, Yes], [6, 13, 8, 8, Yes], [6, 14, 7, 7, Yes], [6, 15, 6, 6, Yes], [6, 16, 5, 5, Yes], [6, 17, 4, 4, Yes], [6, 18, 3, 3, Yes], [6, 19, 2, 2, Yes], [6, 20, 1, 1, Yes], [6, 21, 0, 0, Yes], [6, 22, 26, 26, Yes], [6, 23, 25, 25, Yes], [6, 24, 24, 24, Yes], [6, 25, 23, 23, Yes], [6, 26, 22, 22, Yes], [7, 0, 20, 20, Yes], [7, 1, 19, 19, Yes], [7, 2, 18, 18, Yes], [7, 3, 17, 17, Yes], [7, 4, 16, 16, Yes], [7, 5, 15, 15, Yes], [7, 6, 14, 14, Yes], [7, 7, 13, 13, Yes], [7, 8, 12, 12, Yes], [7, 9, 11, 11, Yes], [7, 10, 10, 10, Yes], [7, 11, 9, 9, Yes], [7, 12, 8, 8, Yes], [7, 13, 7, 7, Yes], [7, 14, 6, 6, Yes], [7, 15, 5, 5, Yes], [7, 16, 4, 4, Yes], [7, 17, 3, 3, Yes], [7, 18, 2, 2, Yes], [7, 19, 1, 1, Yes], [7, 20, 0, 0, Yes], [7, 21, 26, 26, Yes], [7, 22, 25, 25, Yes], [7, 23, 24, 24, Yes], [7, 24, 23, 23, Yes], [7, 25, 22, 22, Yes], [7, 26, 21, 21, Yes], [8, 0, 19, 19, Yes], [8, 1, 18, 18, Yes], [8, 2, 17, 17, Yes], [8, 3, 16, 16, Yes], [8, 4, 15, 15, Yes], [8, 5, 14, 14, Yes], [8, 6, 13, 13, Yes], [8, 7, 12, 12, Yes], [8, 8, 11, 11, Yes], [8, 9, 10, 10, Yes], [8, 10, 9, 9, Yes], [8, 11, 8, 8, Yes], [8, 12, 7, 7, Yes], [8, 13, 6, 6, Yes], [8, 14, 5, 5, Yes], [8, 15, 4, 4, Yes], [8, 16, 3, 3, Yes], [8, 17, 2, 2, Yes], [8, 18, 1, 1, Yes], [8, 19, 0, 0, Yes], [8, 20, 26, 26, Yes], [8, 21, 25, 25, Yes], [8, 22, 24, 24, Yes], [8, 23, 23, 23, Yes], [8, 24, 22, 22, Yes], [8, 25, 21, 21, Yes], [8, 26, 20, 20, Yes], [9, 0, 18, 18, Yes], [9, 1, 17, 17, Yes], [9, 2, 16, 16, Yes], [9, 3, 15, 15, Yes], [9, 4, 14, 14, Yes], [9, 5, 13, 13, Yes], [9, 6, 12, 12, Yes], [9, 7, 11, 11, Yes], [9, 8, 10, 10, Yes], [9, 9, 9, 9, Yes], [9, 10, 8, 8, Yes], [9, 11, 7, 7, Yes], [9, 12, 6, 6, Yes], [9, 13, 5, 5, Yes], [9, 14, 4, 4, Yes], [9, 15, 3, 3, Yes], [9, 16, 2, 2, Yes], [9, 17, 1, 1, Yes], [9, 18, 0, 0, Yes], [9, 19, 26, 26, Yes], [9, 20, 25, 25, Yes], [9, 21, 24, 24, Yes], [9, 22, 23, 23, Yes], [9, 23, 22, 22, Yes], [9, 24, 21, 21, Yes], [9, 25, 20, 20, Yes], [9, 26, 19, 19, Yes], [10, 0, 17, 17, Yes], [10, 1, 16, 16, Yes], [10, 2, 15, 15, Yes], [10, 3, 14, 14, Yes], [10, 4, 13, 13, Yes], [10, 5, 12, 12, Yes], [10, 6, 11, 11, Yes], [10, 7, 10, 10, Yes], [10, 8, 9, 9, Yes], [10, 9, 8, 8, Yes], [10, 10, 7, 7, Yes], [10, 11, 6, 6, Yes], [10, 12, 5, 5, Yes], [10, 13, 4, 4, Yes], [10, 14, 3, 3, Yes], [10, 15, 2, 2, Yes], [10, 16, 1, 1, Yes], [10, 17, 0, 0, Yes], [10, 18, 26, 26, Yes], [10, 19, 25, 25, Yes], [10, 20, 24, 24, Yes], [10, 21, 23, 23, Yes], [10, 22, 22, 22, Yes], [10, 23, 21, 21, Yes], [10, 24, 20, 20, Yes], [10, 25, 19, 19, Yes], [10, 26, 18, 18, Yes], [11, 0, 16, 16, Yes], [11, 1, 15, 15, Yes], [11, 2, 14, 14, Yes], [11, 3, 13, 13, Yes], [11, 4, 12, 12, Yes], [11, 5, 11, 11, Yes], [11, 6, 10, 10, Yes], [11, 7, 9, 9, Yes], [11, 8, 8, 8, Yes], [11, 9, 7, 7, Yes], [11, 10, 6, 6, Yes], [11, 11, 5, 5, Yes], [11, 12, 4, 4, Yes], [11, 13, 3, 3, Yes], [11, 14, 2, 2, Yes], [11, 15, 1, 1, Yes], [11, 16, 0, 0, Yes], [11, 17, 26, 26, Yes], [11, 18, 25, 25, Yes], [11, 19, 24, 24, Yes], [11, 20, 23, 23, Yes], [11, 21, 22, 22, Yes], [11, 22, 21, 21, Yes], [11, 23, 20, 20, Yes], [11, 24, 19, 19, Yes], [11, 25, 18, 18, Yes], [11, 26, 17, 17, Yes], [12, 0, 15, 15, Yes], [12, 1, 14, 14, Yes], [12, 2, 13, 13, Yes], [12, 3, 12, 12, Yes], [12, 4, 11, 11, Yes], [12, 5, 10, 10, Yes], [12, 6, 9, 9, Yes], [12, 7, 8, 8, Yes], [12, 8, 7, 7, Yes], [12, 9, 6, 6, Yes], [12, 10, 5, 5, Yes], [12, 11, 4, 4, Yes], [12, 12, 3, 3, Yes], [12, 13, 2, 2, Yes], [12, 14, 1, 1, Yes], [12, 15, 0, 0, Yes], [12, 16, 26, 26, Yes], [12, 17, 25, 25, Yes], [12, 18, 24, 24, Yes], [12, 19, 23, 23, Yes], [12, 20, 22, 22, Yes], [12, 21, 21, 21, Yes], [12, 22, 20, 20, Yes], [12, 23, 19, 19, Yes], [12, 24, 18, 18, Yes], [12, 25, 17, 17, Yes], [12, 26, 16, 16, Yes], [13, 0, 14, 14, Yes], [13, 1, 13, 13, Yes], [13, 2, 12, 12, Yes], [13, 3, 11, 11, Yes], [13, 4, 10, 10, Yes], [13, 5, 9, 9, Yes], [13, 6, 8, 8, Yes], [13, 7, 7, 7, Yes], [13, 8, 6, 6, Yes], [13, 9, 5, 5, Yes], [13, 10, 4, 4, Yes], [13, 11, 3, 3, Yes], [13, 12, 2, 2, Yes], [13, 13, 1, 1, Yes], [13, 14, 0, 0, Yes], [13, 15, 26, 26, Yes], [13, 16, 25, 25, Yes], [13, 17, 24, 24, Yes], [13, 18, 23, 23, Yes], [13, 19, 22, 22, Yes], [13, 20, 21, 21, Yes], [13, 21, 20, 20, Yes], [13, 22, 19, 19, Yes], [13, 23, 18, 18, Yes], [13, 24, 17, 17, Yes], [13, 25, 16, 16, Yes], [13, 26, 15, 15, Yes], [14, 0, 13, 13, Yes], [14, 1, 12, 12, Yes], [14, 2, 11, 11, Yes], [14, 3, 10, 10, Yes], [14, 4, 9, 9, Yes], [14, 5, 8, 8, Yes], [14, 6, 7, 7, Yes], [14, 7, 6, 6, Yes], [14, 8, 5, 5, Yes], [14, 9, 4, 4, Yes], [14, 10, 3, 3, Yes], [14, 11, 2, 2, Yes], [14, 12, 1, 1, Yes], [14, 13, 0, 0, Yes], [14, 14, 26, 26, Yes], [14, 15, 25, 25, Yes], [14, 16, 24, 24, Yes], [14, 17, 23, 23, Yes], [14, 18, 22, 22, Yes], [14, 19, 21, 21, Yes], [14, 20, 20, 20, Yes], [14, 21, 19, 19, Yes], [14, 22, 18, 18, Yes], [14, 23, 17, 17, Yes], [14, 24, 16, 16, Yes], [14, 25, 15, 15, Yes], [14, 26, 14, 14, Yes], [15, 0, 12, 12, Yes], [15, 1, 11, 11, Yes], [15, 2, 10, 10, Yes], [15, 3, 9, 9, Yes], [15, 4, 8, 8, Yes], [15, 5, 7, 7, Yes], [15, 6, 6, 6, Yes], [15, 7, 5, 5, Yes], [15, 8, 4, 4, Yes], [15, 9, 3, 3, Yes], [15, 10, 2, 2, Yes], [15, 11, 1, 1, Yes], [15, 12, 0, 0, Yes], [15, 13, 26, 26, Yes], [15, 14, 25, 25, Yes], [15, 15, 24, 24, Yes], [15, 16, 23, 23, Yes], [15, 17, 22, 22, Yes], [15, 18, 21, 21, Yes], [15, 19, 20, 20, Yes], [15, 20, 19, 19, Yes], [15, 21, 18, 18, Yes], [15, 22, 17, 17, Yes], [15, 23, 16, 16, Yes], [15, 24, 15, 15, Yes], [15, 25, 14, 14, Yes], [15, 26, 13, 13, Yes], [16, 0, 11, 11, Yes], [16, 1, 10, 10, Yes], [16, 2, 9, 9, Yes], [16, 3, 8, 8, Yes], [16, 4, 7, 7, Yes], [16, 5, 6, 6, Yes], [16, 6, 5, 5, Yes], [16, 7, 4, 4, Yes], [16, 8, 3, 3, Yes], [16, 9, 2, 2, Yes], [16, 10, 1, 1, Yes], [16, 11, 0, 0, Yes], [16, 12, 26, 26, Yes], [16, 13, 25, 25, Yes], [16, 14, 24, 24, Yes], [16, 15, 23, 23, Yes], [16, 16, 22, 22, Yes], [16, 17, 21, 21, Yes], [16, 18, 20, 20, Yes], [16, 19, 19, 19, Yes], [16, 20, 18, 18, Yes], [16, 21, 17, 17, Yes], [16, 22, 16, 16, Yes], [16, 23, 15, 15, Yes], [16, 24, 14, 14, Yes], [16, 25, 13, 13, Yes], [16, 26, 12, 12, Yes], [17, 0, 10, 10, Yes], [17, 1, 9, 9, Yes], [17, 2, 8, 8, Yes], [17, 3, 7, 7, Yes], [17, 4, 6, 6, Yes], [17, 5, 5, 5, Yes], [17, 6, 4, 4, Yes], [17, 7, 3, 3, Yes], [17, 8, 2, 2, Yes], [17, 9, 1, 1, Yes], [17, 10, 0, 0, Yes], [17, 11, 26, 26, Yes], [17, 12, 25, 25, Yes], [17, 13, 24, 24, Yes], [17, 14, 23, 23, Yes], [17, 15, 22, 22, Yes], [17, 16, 21, 21, Yes], [17, 17, 20, 20, Yes], [17, 18, 19, 19, Yes], [17, 19, 18, 18, Yes], [17, 20, 17, 17, Yes], [17, 21, 16, 16, Yes], [17, 22, 15, 15, Yes], [17, 23, 14, 14, Yes], [17, 24, 13, 13, Yes], [17, 25, 12, 12, Yes], [17, 26, 11, 11, Yes], [18, 0, 9, 9, Yes], [18, 1, 8, 8, Yes], [18, 2, 7, 7, Yes], [18, 3, 6, 6, Yes], [18, 4, 5, 5, Yes], [18, 5, 4, 4, Yes], [18, 6, 3, 3, Yes], [18, 7, 2, 2, Yes], [18, 8, 1, 1, Yes], [18, 9, 0, 0, Yes], [18, 10, 26, 26, Yes], [18, 11, 25, 25, Yes], [18, 12, 24, 24, Yes], [18, 13, 23, 23, Yes], [18, 14, 22, 22, Yes], [18, 15, 21, 21, Yes], [18, 16, 20, 20, Yes], [18, 17, 19, 19, Yes], [18, 18, 18, 18, Yes], [18, 19, 17, 17, Yes], [18, 20, 16, 16, Yes], [18, 21, 15, 15, Yes], [18, 22, 14, 14, Yes], [18, 23, 13, 13, Yes], [18, 24, 12, 12, Yes], [18, 25, 11, 11, Yes], [18, 26, 10, 10, Yes], [19, 0, 8, 8, Yes], [19, 1, 7, 7, Yes], [19, 2, 6, 6, Yes], [19, 3, 5, 5, Yes], [19, 4, 4, 4, Yes], [19, 5, 3, 3, Yes], [19, 6, 2, 2, Yes], [19, 7, 1, 1, Yes], [19, 8, 0, 0, Yes], [19, 9, 26, 26, Yes], [19, 10, 25, 25, Yes], [19, 11, 24, 24, Yes], [19, 12, 23, 23, Yes], [19, 13, 22, 22, Yes], [19, 14, 21, 21, Yes], [19, 15, 20, 20, Yes], [19, 16, 19, 19, Yes], [19, 17, 18, 18, Yes], [19, 18, 17, 17, Yes], [19, 19, 16, 16, Yes], [19, 20, 15, 15, Yes], [19, 21, 14, 14, Yes], [19, 22, 13, 13, Yes], [19, 23, 12, 12, Yes], [19, 24, 11, 11, Yes], [19, 25, 10, 10, Yes], [19, 26, 9, 9, Yes], [20, 0, 7, 7, Yes], [20, 1, 6, 6, Yes], [20, 2, 5, 5, Yes], [20, 3, 4, 4, Yes], [20, 4, 3, 3, Yes], [20, 5, 2, 2, Yes], [20, 6, 1, 1, Yes], [20, 7, 0, 0, Yes], [20, 8, 26, 26, Yes], [20, 9, 25, 25, Yes], [20, 10, 24, 24, Yes], [20, 11, 23, 23, Yes], [20, 12, 22, 22, Yes], [20, 13, 21, 21, Yes], [20, 14, 20, 20, Yes], [20, 15, 19, 19, Yes], [20, 16, 18, 18, Yes], [20, 17, 17, 17, Yes], [20, 18, 16, 16, Yes], [20, 19, 15, 15, Yes], [20, 20, 14, 14, Yes], [20, 21, 13, 13, Yes], [20, 22, 12, 12, Yes], [20, 23, 11, 11, Yes], [20, 24, 10, 10, Yes], [20, 25, 9, 9, Yes], [20, 26, 8, 8, Yes], [21, 0, 6, 6, Yes], [21, 1, 5, 5, Yes], [21, 2, 4, 4, Yes], [21, 3, 3, 3, Yes], [21, 4, 2, 2, Yes], [21, 5, 1, 1, Yes], [21, 6, 0, 0, Yes], [21, 7, 26, 26, Yes], [21, 8, 25, 25, Yes], [21, 9, 24, 24, Yes], [21, 10, 23, 23, Yes], [21, 11, 22, 22, Yes], [21, 12, 21, 21, Yes], [21, 13, 20, 20, Yes], [21, 14, 19, 19, Yes], [21, 15, 18, 18, Yes], [21, 16, 17, 17, Yes], [21, 17, 16, 16, Yes], [21, 18, 15, 15, Yes], [21, 19, 14, 14, Yes], [21, 20, 13, 13, Yes], [21, 21, 12, 12, Yes], [21, 22, 11, 11, Yes], [21, 23, 10, 10, Yes], [21, 24, 9, 9, Yes], [21, 25, 8, 8, Yes], [21, 26, 7, 7, Yes], [22, 0, 5, 5, Yes], [22, 1, 4, 4, Yes], [22, 2, 3, 3, Yes], [22, 3, 2, 2, Yes], [22, 4, 1, 1, Yes], [22, 5, 0, 0, Yes], [22, 6, 26, 26, Yes], [22, 7, 25, 25, Yes], [22, 8, 24, 24, Yes], [22, 9, 23, 23, Yes], [22, 10, 22, 22, Yes], [22, 11, 21, 21, Yes], [22, 12, 20, 20, Yes], [22, 13, 19, 19, Yes], [22, 14, 18, 18, Yes], [22, 15, 17, 17, Yes], [22, 16, 16, 16, Yes], [22, 17, 15, 15, Yes], [22, 18, 14, 14, Yes], [22, 19, 13, 13, Yes], [22, 20, 12, 12, Yes], [22, 21, 11, 11, Yes], [22, 22, 10, 10, Yes], [22, 23, 9, 9, Yes], [22, 24, 8, 8, Yes], [22, 25, 7, 7, Yes], [22, 26, 6, 6, Yes], [23, 0, 4, 4, Yes], [23, 1, 3, 3, Yes], [23, 2, 2, 2, Yes], [23, 3, 1, 1, Yes], [23, 4, 0, 0, Yes], [23, 5, 26, 26, Yes], [23, 6, 25, 25, Yes], [23, 7, 24, 24, Yes], [23, 8, 23, 23, Yes], [23, 9, 22, 22, Yes], [23, 10, 21, 21, Yes], [23, 11, 20, 20, Yes], [23, 12, 19, 19, Yes], [23, 13, 18, 18, Yes], [23, 14, 17, 17, Yes], [23, 15, 16, 16, Yes], [23, 16, 15, 15, Yes], [23, 17, 14, 14, Yes], [23, 18, 13, 13, Yes], [23, 19, 12, 12, Yes], [23, 20, 11, 11, Yes], [23, 21, 10, 10, Yes], [23, 22, 9, 9, Yes], [23, 23, 8, 8, Yes], [23, 24, 7, 7, Yes], [23, 25, 6, 6, Yes], [23, 26, 5, 5, Yes], [24, 0, 3, 3, Yes], [24, 1, 2, 2, Yes], [24, 2, 1, 1, Yes], [24, 3, 0, 0, Yes], [24, 4, 26, 26, Yes], [24, 5, 25, 25, Yes], [24, 6, 24, 24, Yes], [24, 7, 23, 23, Yes], [24, 8, 22, 22, Yes], [24, 9, 21, 21, Yes], [24, 10, 20, 20, Yes], [24, 11, 19, 19, Yes], [24, 12, 18, 18, Yes], [24, 13, 17, 17, Yes], [24, 14, 16, 16, Yes], [24, 15, 15, 15, Yes], [24, 16, 14, 14, Yes], [24, 17, 13, 13, Yes], [24, 18, 12, 12, Yes], [24, 19, 11, 11, Yes], [24, 20, 10, 10, Yes], [24, 21, 9, 9, Yes], [24, 22, 8, 8, Yes], [24, 23, 7, 7, Yes], [24, 24, 6, 6, Yes], [24, 25, 5, 5, Yes], [24, 26, 4, 4, Yes], [25, 0, 2, 2, Yes], [25, 1, 1, 1, Yes], [25, 2, 0, 0, Yes], [25, 3, 26, 26, Yes], [25, 4, 25, 25, Yes], [25, 5, 24, 24, Yes], [25, 6, 23, 23, Yes], [25, 7, 22, 22, Yes], [25, 8, 21, 21, Yes], [25, 9, 20, 20, Yes], [25, 10, 19, 19, Yes], [25, 11, 18, 18, Yes], [25, 12, 17, 17, Yes], [25, 13, 16, 16, Yes], [25, 14, 15, 15, Yes], [25, 15, 14, 14, Yes], [25, 16, 13, 13, Yes], [25, 17, 12, 12, Yes], [25, 18, 11, 11, Yes], [25, 19, 10, 10, Yes], [25, 20, 9, 9, Yes], [25, 21, 8, 8, Yes], [25, 22, 7, 7, Yes], [25, 23, 6, 6, Yes], [25, 24, 5, 5, Yes], [25, 25, 4, 4, Yes], [25, 26, 3, 3, Yes], [26, 0, 1, 1, Yes], [26, 1, 0, 0, Yes], [26, 2, 26, 26, Yes], [26, 3, 25, 25, Yes], [26, 4, 24, 24, Yes], [26, 5, 23, 23, Yes], [26, 6, 22, 22, Yes], [26, 7, 21, 21, Yes], [26, 8, 20, 20, Yes], [26, 9, 19, 19, Yes], [26, 10, 18, 18, Yes], [26, 11, 17, 17, Yes], [26, 12, 16, 16, Yes], [26, 13, 15, 15, Yes], [26, 14, 14, 14, Yes], [26, 15, 13, 13, Yes], [26, 16, 12, 12, Yes], [26, 17, 11, 11, Yes], [26, 18, 10, 10, Yes], [26, 19, 9, 9, Yes], [26, 20, 8, 8, Yes], [26, 21, 7, 7, Yes], [26, 22, 6, 6, Yes], [26, 23, 5, 5, Yes], [26, 24, 4, 4, Yes], [26, 25, 3, 3, Yes], [26, 26, 2, 2, Yes]], Yes, null, complete_total_map_satisfies_every_product_pair).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-homomorphism-qualification · candidate
26. The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup
Use complete source Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).
View answer and explanation
([0, 2], Yes, Yes, Yes, [[0, [0, 2], Yes], [1, [0, 2], Yes], [2, [0, 2], Yes], [3, [0, 2], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)
Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2], Yes, Yes, Yes, [[0, [0, 2], Yes], [1, [0, 2], Yes], [2, [0, 2], Yes], [3, [0, 2], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-normal-subgroup-audit · candidate
27. The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).
View answer and explanation
([0, 2, 4], Yes, Yes, Yes, [[0, [0, 2, 4], Yes], [1, [0, 2, 4], Yes], [2, [0, 2, 4], Yes], [3, [0, 2, 4], Yes], [4, [0, 2, 4], Yes], [5, [0, 2, 4], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)
Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4], Yes, Yes, Yes, [[0, [0, 2, 4], Yes], [1, [0, 2, 4], Yes], [2, [0, 2, 4], Yes], [3, [0, 2, 4], Yes], [4, [0, 2, 4], Yes], [5, [0, 2, 4], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-normal-subgroup-audit · candidate
28. The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3], [5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).
View answer and explanation
([0, 2, 4, 6], Yes, Yes, Yes, [[0, [0, 2, 4, 6], Yes], [1, [0, 2, 4, 6], Yes], [2, [0, 2, 4, 6], Yes], [3, [0, 2, 4, 6], Yes], [4, [0, 2, 4, 6], Yes], [5, [0, 2, 4, 6], Yes], [6, [0, 2, 4, 6], Yes], [7, [0, 2, 4, 6], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)
Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6], Yes, Yes, Yes, [[0, [0, 2, 4, 6], Yes], [1, [0, 2, 4, 6], Yes], [2, [0, 2, 4, 6], Yes], [3, [0, 2, 4, 6], Yes], [4, [0, 2, 4, 6], Yes], [5, [0, 2, 4, 6], Yes], [6, [0, 2, 4, 6], Yes], [7, [0, 2, 4, 6], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-normal-subgroup-audit · candidate
29. The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).
View answer and explanation
([0, 2, 4, 6, 8], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8], Yes], [1, [0, 2, 4, 6, 8], Yes], [2, [0, 2, 4, 6, 8], Yes], [3, [0, 2, 4, 6, 8], Yes], [4, [0, 2, 4, 6, 8], Yes], [5, [0, 2, 4, 6, 8], Yes], [6, [0, 2, 4, 6, 8], Yes], [7, [0, 2, 4, 6, 8], Yes], [8, [0, 2, 4, 6, 8], Yes], [9, [0, 2, 4, 6, 8], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)
Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8], Yes], [1, [0, 2, 4, 6, 8], Yes], [2, [0, 2, 4, 6, 8], Yes], [3, [0, 2, 4, 6, 8], Yes], [4, [0, 2, 4, 6, 8], Yes], [5, [0, 2, 4, 6, 8], Yes], [6, [0, 2, 4, 6, 8], Yes], [7, [0, 2, 4, 6, 8], Yes], [8, [0, 2, 4, 6, 8], Yes], [9, [0, 2, 4, 6, 8], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-normal-subgroup-audit · candidate
30. The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).
View answer and explanation
([0, 2, 4, 6, 8, 10], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10], Yes], [1, [0, 2, 4, 6, 8, 10], Yes], [2, [0, 2, 4, 6, 8, 10], Yes], [3, [0, 2, 4, 6, 8, 10], Yes], [4, [0, 2, 4, 6, 8, 10], Yes], [5, [0, 2, 4, 6, 8, 10], Yes], [6, [0, 2, 4, 6, 8, 10], Yes], [7, [0, 2, 4, 6, 8, 10], Yes], [8, [0, 2, 4, 6, 8, 10], Yes], [9, [0, 2, 4, 6, 8, 10], Yes], [10, [0, 2, 4, 6, 8, 10], Yes], [11, [0, 2, 4, 6, 8, 10], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)
Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10], Yes], [1, [0, 2, 4, 6, 8, 10], Yes], [2, [0, 2, 4, 6, 8, 10], Yes], [3, [0, 2, 4, 6, 8, 10], Yes], [4, [0, 2, 4, 6, 8, 10], Yes], [5, [0, 2, 4, 6, 8, 10], Yes], [6, [0, 2, 4, 6, 8, 10], Yes], [7, [0, 2, 4, 6, 8, 10], Yes], [8, [0, 2, 4, 6, 8, 10], Yes], [9, [0, 2, 4, 6, 8, 10], Yes], [10, [0, 2, 4, 6, 8, 10], Yes], [11, [0, 2, 4, 6, 8, 10], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-normal-subgroup-audit · candidate
31. The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).
View answer and explanation
([0, 2, 4, 6, 8, 10, 12], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12], Yes], [1, [0, 2, 4, 6, 8, 10, 12], Yes], [2, [0, 2, 4, 6, 8, 10, 12], Yes], [3, [0, 2, 4, 6, 8, 10, 12], Yes], [4, [0, 2, 4, 6, 8, 10, 12], Yes], [5, [0, 2, 4, 6, 8, 10, 12], Yes], [6, [0, 2, 4, 6, 8, 10, 12], Yes], [7, [0, 2, 4, 6, 8, 10, 12], Yes], [8, [0, 2, 4, 6, 8, 10, 12], Yes], [9, [0, 2, 4, 6, 8, 10, 12], Yes], [10, [0, 2, 4, 6, 8, 10, 12], Yes], [11, [0, 2, 4, 6, 8, 10, 12], Yes], [12, [0, 2, 4, 6, 8, 10, 12], Yes], [13, [0, 2, 4, 6, 8, 10, 12], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)
Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12], Yes], [1, [0, 2, 4, 6, 8, 10, 12], Yes], [2, [0, 2, 4, 6, 8, 10, 12], Yes], [3, [0, 2, 4, 6, 8, 10, 12], Yes], [4, [0, 2, 4, 6, 8, 10, 12], Yes], [5, [0, 2, 4, 6, 8, 10, 12], Yes], [6, [0, 2, 4, 6, 8, 10, 12], Yes], [7, [0, 2, 4, 6, 8, 10, 12], Yes], [8, [0, 2, 4, 6, 8, 10, 12], Yes], [9, [0, 2, 4, 6, 8, 10, 12], Yes], [10, [0, 2, 4, 6, 8, 10, 12], Yes], [11, [0, 2, 4, 6, 8, 10, 12], Yes], [12, [0, 2, 4, 6, 8, 10, 12], Yes], [13, [0, 2, 4, 6, 8, 10, 12], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-normal-subgroup-audit · candidate
32. The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)
Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-normal-subgroup-audit · candidate
33. The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)
Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-normal-subgroup-audit · candidate
34. The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)
Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-normal-subgroup-audit · candidate
35. The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)
Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-normal-subgroup-audit · candidate
36. The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)
Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-normal-subgroup-audit · candidate
37. The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)
Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-normal-subgroup-audit · candidate
38. The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)
Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-normal-subgroup-audit · candidate
39. The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)
Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-normal-subgroup-audit · candidate
40. The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)
Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-normal-subgroup-audit · candidate
41. The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)
Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-normal-subgroup-audit · candidate
42. The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)
Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-normal-subgroup-audit · candidate
43. The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)
Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-normal-subgroup-audit · candidate
44. The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)
Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-normal-subgroup-audit · candidate
45. The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)
Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-normal-subgroup-audit · candidate
46. The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [42, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [43, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)
Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [42, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes], [43, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-normal-subgroup-audit · candidate
47. The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [42, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [43, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [44, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [45, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)
Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [42, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [43, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [44, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes], [45, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-normal-subgroup-audit · candidate
48. The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [42, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [43, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [44, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [45, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [46, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [47, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)
Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [42, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [43, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [44, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [45, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [46, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [47, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-normal-subgroup-audit · candidate
49. The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [42, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [43, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [44, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [45, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [46, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [47, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [48, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [49, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)
Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [42, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [43, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [44, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [45, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [46, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [47, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [48, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes], [49, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-normal-subgroup-audit · candidate
50. The Identity Preimage of a Qualified Homomorphism Is a Normal Subgroup
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).
View answer and explanation
([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [42, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [43, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [44, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [45, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [46, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [47, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [48, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [49, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [50, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [51, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)
Require the full homomorphism law. Enumerate the entire preimage of the target identity, then audit source identity/product/inverse closure and every g*K*g^(-1). Keep subgroup and normality flags distinct. This finite audit does not assert a quotient/image isomorphism. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [42, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [43, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [44, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [45, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [46, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [47, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [48, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [49, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [50, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes], [51, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-normal-subgroup-audit · candidate
51. Image Equality Agrees with a Inverse Times b Membership in the Kernel
Use complete source Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=0, b=2. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).
equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The final flag compares truth values; it does not assert f(a)=f(b).
View answer and explanation
(0, 2, 0, 0, Yes, 2, Yes, [0, 2], [0, 2], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)
Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (0, 2, 0, 0, Yes, 2, Yes, [0, 2], [0, 2], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-fiber-coset-equivalence · candidate
52. Image Equality Agrees with a Inverse Times b Membership in the Kernel
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=1, b=3. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).
equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The final flag compares truth values; it does not assert f(a)=f(b).
View answer and explanation
(1, 3, 1, 1, Yes, 2, Yes, [1, 3, 5], [1, 3, 5], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)
Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (1, 3, 1, 1, Yes, 2, Yes, [1, 3, 5], [1, 3, 5], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-fiber-coset-equivalence · candidate
53. Image Equality Agrees with a Inverse Times b Membership in the Kernel
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3], [5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=2, b=4. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).
equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The final flag compares truth values; it does not assert f(a)=f(b).
View answer and explanation
(2, 4, 0, 0, Yes, 2, Yes, [0, 2, 4, 6], [0, 2, 4, 6], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)
Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (2, 4, 0, 0, Yes, 2, Yes, [0, 2, 4, 6], [0, 2, 4, 6], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-fiber-coset-equivalence · candidate
54. Image Equality Agrees with a Inverse Times b Membership in the Kernel
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=3, b=5. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).
equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The final flag compares truth values; it does not assert f(a)=f(b).
View answer and explanation
(3, 5, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9], [1, 3, 5, 7, 9], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)
Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (3, 5, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9], [1, 3, 5, 7, 9], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-fiber-coset-equivalence · candidate
55. Image Equality Agrees with a Inverse Times b Membership in the Kernel
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=4, b=6. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).
equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The final flag compares truth values; it does not assert f(a)=f(b).
View answer and explanation
(4, 6, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10], [0, 2, 4, 6, 8, 10], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)
Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (4, 6, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10], [0, 2, 4, 6, 8, 10], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-fiber-coset-equivalence · candidate
56. Image Equality Agrees with a Inverse Times b Membership in the Kernel
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=5, b=7. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).
equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The final flag compares truth values; it does not assert f(a)=f(b).
View answer and explanation
(5, 7, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13], [1, 3, 5, 7, 9, 11, 13], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)
Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (5, 7, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13], [1, 3, 5, 7, 9, 11, 13], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-fiber-coset-equivalence · candidate
57. Image Equality Agrees with a Inverse Times b Membership in the Kernel
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=6, b=8. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).
equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The final flag compares truth values; it does not assert f(a)=f(b).
View answer and explanation
(6, 8, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14], [0, 2, 4, 6, 8, 10, 12, 14], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)
Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (6, 8, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14], [0, 2, 4, 6, 8, 10, 12, 14], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-fiber-coset-equivalence · candidate
58. Image Equality Agrees with a Inverse Times b Membership in the Kernel
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=7, b=9. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).
equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The final flag compares truth values; it does not assert f(a)=f(b).
View answer and explanation
(7, 9, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17], [1, 3, 5, 7, 9, 11, 13, 15, 17], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)
Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (7, 9, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17], [1, 3, 5, 7, 9, 11, 13, 15, 17], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-fiber-coset-equivalence · candidate
59. Image Equality Agrees with a Inverse Times b Membership in the Kernel
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=8, b=10. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).
equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The final flag compares truth values; it does not assert f(a)=f(b).
View answer and explanation
(8, 10, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)
Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (8, 10, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-fiber-coset-equivalence · candidate
60. Image Equality Agrees with a Inverse Times b Membership in the Kernel
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=9, b=11. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).
equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The final flag compares truth values; it does not assert f(a)=f(b).
View answer and explanation
(9, 11, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)
Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (9, 11, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-fiber-coset-equivalence · candidate
61. Image Equality Agrees with a Inverse Times b Membership in the Kernel
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=10, b=12. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).
equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The final flag compares truth values; it does not assert f(a)=f(b).
View answer and explanation
(10, 12, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)
Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (10, 12, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-fiber-coset-equivalence · candidate
62. Image Equality Agrees with a Inverse Times b Membership in the Kernel
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=11, b=13. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).
equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The final flag compares truth values; it does not assert f(a)=f(b).
View answer and explanation
(11, 13, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)
Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (11, 13, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-fiber-coset-equivalence · candidate
63. Image Equality Agrees with a Inverse Times b Membership in the Kernel
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=12, b=14. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).
equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The final flag compares truth values; it does not assert f(a)=f(b).
View answer and explanation
(12, 14, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)
Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (12, 14, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-fiber-coset-equivalence · candidate
64. Image Equality Agrees with a Inverse Times b Membership in the Kernel
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=13, b=15. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).
equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The final flag compares truth values; it does not assert f(a)=f(b).
View answer and explanation
(13, 15, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)
Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (13, 15, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-fiber-coset-equivalence · candidate
65. Image Equality Agrees with a Inverse Times b Membership in the Kernel
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=14, b=16. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).
equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The final flag compares truth values; it does not assert f(a)=f(b).
View answer and explanation
(14, 16, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)
Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (14, 16, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-fiber-coset-equivalence · candidate
66. Image Equality Agrees with a Inverse Times b Membership in the Kernel
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=15, b=17. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).
equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The final flag compares truth values; it does not assert f(a)=f(b).
View answer and explanation
(15, 17, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)
Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (15, 17, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-fiber-coset-equivalence · candidate
67. Image Equality Agrees with a Inverse Times b Membership in the Kernel
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=16, b=18. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).
equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The final flag compares truth values; it does not assert f(a)=f(b).
View answer and explanation
(16, 18, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)
Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (16, 18, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-fiber-coset-equivalence · candidate
68. Image Equality Agrees with a Inverse Times b Membership in the Kernel
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=17, b=19. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).
equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The final flag compares truth values; it does not assert f(a)=f(b).
View answer and explanation
(17, 19, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)
Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (17, 19, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-fiber-coset-equivalence · candidate
69. Image Equality Agrees with a Inverse Times b Membership in the Kernel
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=18, b=20. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).
equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The final flag compares truth values; it does not assert f(a)=f(b).
View answer and explanation
(18, 20, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)
Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (18, 20, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-fiber-coset-equivalence · candidate
70. Image Equality Agrees with a Inverse Times b Membership in the Kernel
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [40, 41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [41, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=19, b=21. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).
equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The final flag compares truth values; it does not assert f(a)=f(b).
View answer and explanation
(19, 21, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)
Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (19, 21, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-fiber-coset-equivalence · candidate
71. Image Equality Agrees with a Inverse Times b Membership in the Kernel
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=20, b=22. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).
equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The final flag compares truth values; it does not assert f(a)=f(b).
View answer and explanation
(20, 22, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)
Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (20, 22, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-fiber-coset-equivalence · candidate
72. Image Equality Agrees with a Inverse Times b Membership in the Kernel
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [40, 41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [41, 42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [42, 43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [43, 44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [44, 45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [45, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=21, b=23. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).
equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The final flag compares truth values; it does not assert f(a)=f(b).
View answer and explanation
(21, 23, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)
Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (21, 23, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-fiber-coset-equivalence · candidate
73. Image Equality Agrees with a Inverse Times b Membership in the Kernel
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [40, 41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [41, 42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [42, 43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [43, 44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [44, 45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [45, 46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [46, 47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [47, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=22, b=24. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).
equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The final flag compares truth values; it does not assert f(a)=f(b).
View answer and explanation
(22, 24, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)
Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (22, 24, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-fiber-coset-equivalence · candidate
74. Image Equality Agrees with a Inverse Times b Membership in the Kernel
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [41, 42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [42, 43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [43, 44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [44, 45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [45, 46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [46, 47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [47, 48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [48, 49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [49, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=23, b=25. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).
equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The final flag compares truth values; it does not assert f(a)=f(b).
View answer and explanation
(23, 25, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)
Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (23, 25, 1, 1, Yes, 2, Yes, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-fiber-coset-equivalence · candidate
75. Image Equality Agrees with a Inverse Times b Membership in the Kernel
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38], [40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], [42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], [43, 44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], [44, 45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], [45, 46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], [46, 47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], [47, 48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], [48, 49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47], [49, 50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48], [50, 51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], [51, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Given source a=24, b=26. Give (a, b, image_a, image_b, equal_images, left_division_a_inverse_b, difference_in_kernel, fiber, left_coset, fiber_equals_coset, equivalence_holds, certificate).
equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The final flag compares truth values; it does not assert f(a)=f(b).
View answer and explanation
(24, 26, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset)
Require homomorphism qualification. Compute a^(-1)*b, its kernel membership, the full fiber of f(a) and left coset a*K. This fiber is nonempty because it contains a; no surjectivity onto the target is assumed. equal_images and difference_in_kernel may both be No while equivalence_holds is Yes. The exact result, retaining the requested order, is (24, 26, 0, 0, Yes, 2, Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50], Yes, Yes, f(a)=f(b)_iff_a_inverse_b_in_kernel_and_nonempty_fiber_is_a_left_coset).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-fiber-coset-equivalence · candidate
76. A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem
Use complete source Cayley table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], source identity=0, target table=[[0, 1, 2], [1, 2, 0], [2, 0, 1]], target identity=0, and total mapping=[0, 1, 2]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).
The fixed No withholds quotient/first-isomorphism claims, not the preceding injectivity or trivial-kernel equivalence.
View answer and explanation
([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)
Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-trivial-injective-boundary · candidate
77. A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem
Use complete source Cayley table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], source identity=0, target table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], target identity=0, and total mapping=[0, 1, 2, 3]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).
The fixed No withholds quotient/first-isomorphism claims, not the preceding injectivity or trivial-kernel equivalence.
View answer and explanation
([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)
Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-trivial-injective-boundary · candidate
78. A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem
Use complete source Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], source identity=0, target table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], target identity=0, and total mapping=[0, 1, 2, 3, 4]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).
The fixed No withholds quotient/first-isomorphism claims, not the preceding injectivity or trivial-kernel equivalence.
View answer and explanation
([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)
Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-trivial-injective-boundary · candidate
79. A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], source identity=0, target table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).
The fixed No withholds quotient/first-isomorphism claims, not the preceding injectivity or trivial-kernel equivalence.
View answer and explanation
([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)
Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-trivial-injective-boundary · candidate
80. A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6, 0], [2, 3, 4, 5, 6, 0, 1], [3, 4, 5, 6, 0, 1, 2], [4, 5, 6, 0, 1, 2, 3], [5, 6, 0, 1, 2, 3, 4], [6, 0, 1, 2, 3, 4, 5]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).
The fixed No withholds quotient/first-isomorphism claims, not the preceding injectivity or trivial-kernel equivalence.
View answer and explanation
([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)
Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-trivial-injective-boundary · candidate
81. A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3], [5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3], [5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).
The fixed No withholds quotient/first-isomorphism claims, not the preceding injectivity or trivial-kernel equivalence.
View answer and explanation
([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)
Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-trivial-injective-boundary · candidate
82. A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 6, 7, 8, 0], [2, 3, 4, 5, 6, 7, 8, 0, 1], [3, 4, 5, 6, 7, 8, 0, 1, 2], [4, 5, 6, 7, 8, 0, 1, 2, 3], [5, 6, 7, 8, 0, 1, 2, 3, 4], [6, 7, 8, 0, 1, 2, 3, 4, 5], [7, 8, 0, 1, 2, 3, 4, 5, 6], [8, 0, 1, 2, 3, 4, 5, 6, 7]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8], [1, 2, 3, 4, 5, 6, 7, 8, 0], [2, 3, 4, 5, 6, 7, 8, 0, 1], [3, 4, 5, 6, 7, 8, 0, 1, 2], [4, 5, 6, 7, 8, 0, 1, 2, 3], [5, 6, 7, 8, 0, 1, 2, 3, 4], [6, 7, 8, 0, 1, 2, 3, 4, 5], [7, 8, 0, 1, 2, 3, 4, 5, 6], [8, 0, 1, 2, 3, 4, 5, 6, 7]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).
The fixed No withholds quotient/first-isomorphism claims, not the preceding injectivity or trivial-kernel equivalence.
View answer and explanation
([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)
Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-trivial-injective-boundary · candidate
83. A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [2, 3, 4, 5, 6, 7, 8, 9, 0, 1], [3, 4, 5, 6, 7, 8, 9, 0, 1, 2], [4, 5, 6, 7, 8, 9, 0, 1, 2, 3], [5, 6, 7, 8, 9, 0, 1, 2, 3, 4], [6, 7, 8, 9, 0, 1, 2, 3, 4, 5], [7, 8, 9, 0, 1, 2, 3, 4, 5, 6], [8, 9, 0, 1, 2, 3, 4, 5, 6, 7], [9, 0, 1, 2, 3, 4, 5, 6, 7, 8]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).
The fixed No withholds quotient/first-isomorphism claims, not the preceding injectivity or trivial-kernel equivalence.
View answer and explanation
([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)
Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-trivial-injective-boundary · candidate
84. A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).
The fixed No withholds quotient/first-isomorphism claims, not the preceding injectivity or trivial-kernel equivalence.
View answer and explanation
([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)
Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-trivial-injective-boundary · candidate
85. A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).
The fixed No withholds quotient/first-isomorphism claims, not the preceding injectivity or trivial-kernel equivalence.
View answer and explanation
([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)
Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-trivial-injective-boundary · candidate
86. A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).
The fixed No withholds quotient/first-isomorphism claims, not the preceding injectivity or trivial-kernel equivalence.
View answer and explanation
([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)
Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-trivial-injective-boundary · candidate
87. A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).
The fixed No withholds quotient/first-isomorphism claims, not the preceding injectivity or trivial-kernel equivalence.
View answer and explanation
([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)
Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-trivial-injective-boundary · candidate
88. A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).
The fixed No withholds quotient/first-isomorphism claims, not the preceding injectivity or trivial-kernel equivalence.
View answer and explanation
([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)
Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-trivial-injective-boundary · candidate
89. A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).
The fixed No withholds quotient/first-isomorphism claims, not the preceding injectivity or trivial-kernel equivalence.
View answer and explanation
([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)
Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-trivial-injective-boundary · candidate
90. A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).
The fixed No withholds quotient/first-isomorphism claims, not the preceding injectivity or trivial-kernel equivalence.
View answer and explanation
([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)
Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-trivial-injective-boundary · candidate
91. A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).
The fixed No withholds quotient/first-isomorphism claims, not the preceding injectivity or trivial-kernel equivalence.
View answer and explanation
([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)
Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-trivial-injective-boundary · candidate
92. A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).
The fixed No withholds quotient/first-isomorphism claims, not the preceding injectivity or trivial-kernel equivalence.
View answer and explanation
([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)
Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-trivial-injective-boundary · candidate
93. A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).
The fixed No withholds quotient/first-isomorphism claims, not the preceding injectivity or trivial-kernel equivalence.
View answer and explanation
([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)
Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-trivial-injective-boundary · candidate
94. A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).
The fixed No withholds quotient/first-isomorphism claims, not the preceding injectivity or trivial-kernel equivalence.
View answer and explanation
([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)
Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-trivial-injective-boundary · candidate
95. A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).
The fixed No withholds quotient/first-isomorphism claims, not the preceding injectivity or trivial-kernel equivalence.
View answer and explanation
([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)
Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-trivial-injective-boundary · candidate
96. A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).
The fixed No withholds quotient/first-isomorphism claims, not the preceding injectivity or trivial-kernel equivalence.
View answer and explanation
([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)
Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-trivial-injective-boundary · candidate
97. A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).
The fixed No withholds quotient/first-isomorphism claims, not the preceding injectivity or trivial-kernel equivalence.
View answer and explanation
([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)
Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-trivial-injective-boundary · candidate
98. A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).
The fixed No withholds quotient/first-isomorphism claims, not the preceding injectivity or trivial-kernel equivalence.
View answer and explanation
([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)
Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-trivial-injective-boundary · candidate
99. A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).
The fixed No withholds quotient/first-isomorphism claims, not the preceding injectivity or trivial-kernel equivalence.
View answer and explanation
([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)
Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-trivial-injective-boundary · candidate
100. A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem
Use complete source Cayley table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]], source identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]], target identity=0, and total mapping=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary).
The fixed No withholds quotient/first-isomorphism claims, not the preceding injectivity or trivial-kernel equivalence.
View answer and explanation
([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)
Require a homomorphism and compare injectivity with the kernel being exactly the source-identity singleton. The fixed No withholds quotient/first-isomorphism claims; it does not negate injectivity or the equivalence flag. The exact result, retaining the requested order, is ([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim).
Reviewed step-by-step derivation
Kernel of a finite group homomorphism · qf-mod45-kernel-trivial-injective-boundary · candidate
Canonical question JSONL · English JSONL · Validation report