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Question 24986: A Trivial Kernel Is Equivalent to Injectivity, Not Yet the First Isomorphism Theorem

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

R0048-24986 · q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0086

Original question and givens

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).

  • G001 — 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).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

source roles = [kernel-trivial-injective-boundary]

After

derived = [[kernel, injective, trivial_kernel, equivalence_holds, quotient_or_first_isomorphism_claim, boundary]]

Conditions: none

List the full source field order without JavaScript,keeping original data,Boolean flags,mathematical objects and nonclaims distinct.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · implies

Before

source roles = [Kernel ORIGINAL SOURCE group, 64, [[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]], 0]

After

derived = [13, [[0, 0, 0, 0], [1, 1, 1, 1], [2, 2, 2, 2], [3, 3, 3, 3], [4, 4, 4, 4], [5, 5, 5, 5], [6, 6, 6, 6], [7, 7, 7, 7], [8, 8, 8, 8], [9, 9, 9, 9], [10, 10, 10, 10], [11, 11, 11, 11], [12, 12, 12, 12]], Yes]

Conditions: none

Validate the complete closed n-by-n table and actual identity label,not an assumed0.Mod44 allows32 and mod45 allows64;do not mix carriers or tables.

Step identifiers and dependencies

Rule: Group_complete_Cayley_signature_actual_identity · Uses: G001, S001

S003 · implies

Before

source roles = [Kernel ORIGINAL SOURCE group, 13, FOR EVERY x,y,z in0..n-1]

After

derived = [[[Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], 2197, Yes]

Conditions: none

Entry(x,y) checks (xy)z=x(yz) for every z,covering all n^3 triples,not samples.The complete table is already displayed,so do not print millions of redundant lookups.

Step identifiers and dependencies

Rule: Group_ALL_associativity_triples_compact_complete_pair_matrix · Uses: G001, S002

S004 · implies

Before

source roles = [Kernel ORIGINAL SOURCE group, 0]

After

derived = [[0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0], [1, 12, 0, 0], [2, 11, 0, 0], [3, 10, 0, 0], [4, 9, 0, 0], [5, 8, 0, 0], [6, 7, 0, 0], [7, 6, 0, 0], [8, 5, 0, 0], [9, 4, 0, 0], [10, 3, 0, 0], [11, 2, 0, 0], [12, 1, 0, 0]], Yes]

Conditions: none

Every left and right inverse must return the same actual identity;associativity gives uniqueness.This completes the group premises.

Step identifiers and dependencies

Rule: Group_ALL_two_sided_inverses · Uses: G001, S003

S005 · implies

Before

source roles = [Kernel ORIGINAL TARGET group, 64, [[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]], 0]

After

derived = [13, [[0, 0, 0, 0], [1, 1, 1, 1], [2, 2, 2, 2], [3, 3, 3, 3], [4, 4, 4, 4], [5, 5, 5, 5], [6, 6, 6, 6], [7, 7, 7, 7], [8, 8, 8, 8], [9, 9, 9, 9], [10, 10, 10, 10], [11, 11, 11, 11], [12, 12, 12, 12]], Yes]

Conditions: none

Validate the complete closed n-by-n table and actual identity label,not an assumed0.Mod44 allows32 and mod45 allows64;do not mix carriers or tables.

Step identifiers and dependencies

Rule: Group_complete_Cayley_signature_actual_identity · Uses: G001, S004

S006 · implies

Before

source roles = [Kernel ORIGINAL TARGET group, 13, FOR EVERY x,y,z in0..n-1]

After

derived = [[[Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], 2197, Yes]

Conditions: none

Entry(x,y) checks (xy)z=x(yz) for every z,covering all n^3 triples,not samples.The complete table is already displayed,so do not print millions of redundant lookups.

Step identifiers and dependencies

Rule: Group_ALL_associativity_triples_compact_complete_pair_matrix · Uses: G001, S005

S007 · implies

Before

source roles = [Kernel ORIGINAL TARGET group, 0]

After

derived = [[0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [[0, 0, 0, 0], [1, 12, 0, 0], [2, 11, 0, 0], [3, 10, 0, 0], [4, 9, 0, 0], [5, 8, 0, 0], [6, 7, 0, 0], [7, 6, 0, 0], [8, 5, 0, 0], [9, 4, 0, 0], [10, 3, 0, 0], [11, 2, 0, 0], [12, 1, 0, 0]], Yes]

Conditions: none

Every left and right inverse must return the same actual identity;associativity gives uniqueness.This completes the group premises.

Step identifiers and dependencies

Rule: Group_ALL_two_sided_inverses · Uses: G001, S006

S008 · implies

Before

source roles = [[[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]], [[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]], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]]

After

derived = [[[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [0, 2, 2, 2, Yes], [0, 3, 3, 3, Yes], [0, 4, 4, 4, Yes], [0, 5, 5, 5, Yes], [0, 6, 6, 6, Yes], [0, 7, 7, 7, Yes], [0, 8, 8, 8, Yes], [0, 9, 9, 9, Yes], [0, 10, 10, 10, Yes], [0, 11, 11, 11, Yes], [0, 12, 12, 12, Yes], [1, 0, 1, 1, Yes], [1, 1, 2, 2, Yes], [1, 2, 3, 3, Yes], [1, 3, 4, 4, Yes], [1, 4, 5, 5, Yes], [1, 5, 6, 6, Yes], [1, 6, 7, 7, Yes], [1, 7, 8, 8, Yes], [1, 8, 9, 9, Yes], [1, 9, 10, 10, Yes], [1, 10, 11, 11, Yes], [1, 11, 12, 12, Yes], [1, 12, 0, 0, Yes], [2, 0, 2, 2, Yes], [2, 1, 3, 3, Yes], [2, 2, 4, 4, Yes], [2, 3, 5, 5, Yes], [2, 4, 6, 6, Yes], [2, 5, 7, 7, Yes], [2, 6, 8, 8, Yes], [2, 7, 9, 9, Yes], [2, 8, 10, 10, Yes], [2, 9, 11, 11, Yes], [2, 10, 12, 12, Yes], [2, 11, 0, 0, Yes], [2, 12, 1, 1, Yes], [3, 0, 3, 3, Yes], [3, 1, 4, 4, Yes], [3, 2, 5, 5, Yes], [3, 3, 6, 6, Yes], [3, 4, 7, 7, Yes], [3, 5, 8, 8, Yes], [3, 6, 9, 9, Yes], [3, 7, 10, 10, Yes], [3, 8, 11, 11, Yes], [3, 9, 12, 12, Yes], [3, 10, 0, 0, Yes], [3, 11, 1, 1, Yes], [3, 12, 2, 2, Yes], [4, 0, 4, 4, Yes], [4, 1, 5, 5, Yes], [4, 2, 6, 6, Yes], [4, 3, 7, 7, Yes], [4, 4, 8, 8, Yes], [4, 5, 9, 9, Yes], [4, 6, 10, 10, Yes], [4, 7, 11, 11, Yes], [4, 8, 12, 12, Yes], [4, 9, 0, 0, Yes], [4, 10, 1, 1, Yes], [4, 11, 2, 2, Yes], [4, 12, 3, 3, Yes], [5, 0, 5, 5, Yes], [5, 1, 6, 6, Yes], [5, 2, 7, 7, Yes], [5, 3, 8, 8, Yes], [5, 4, 9, 9, Yes], [5, 5, 10, 10, Yes], [5, 6, 11, 11, Yes], [5, 7, 12, 12, Yes], [5, 8, 0, 0, Yes], [5, 9, 1, 1, Yes], [5, 10, 2, 2, Yes], [5, 11, 3, 3, Yes], [5, 12, 4, 4, Yes], [6, 0, 6, 6, Yes], [6, 1, 7, 7, Yes], [6, 2, 8, 8, Yes], [6, 3, 9, 9, Yes], [6, 4, 10, 10, Yes], [6, 5, 11, 11, Yes], [6, 6, 12, 12, Yes], [6, 7, 0, 0, Yes], [6, 8, 1, 1, Yes], [6, 9, 2, 2, Yes], [6, 10, 3, 3, Yes], [6, 11, 4, 4, Yes], [6, 12, 5, 5, Yes], [7, 0, 7, 7, Yes], [7, 1, 8, 8, Yes], [7, 2, 9, 9, Yes], [7, 3, 10, 10, Yes], [7, 4, 11, 11, Yes], [7, 5, 12, 12, Yes], [7, 6, 0, 0, Yes], [7, 7, 1, 1, Yes], [7, 8, 2, 2, Yes], [7, 9, 3, 3, Yes], [7, 10, 4, 4, Yes], [7, 11, 5, 5, Yes], [7, 12, 6, 6, Yes], [8, 0, 8, 8, Yes], [8, 1, 9, 9, Yes], [8, 2, 10, 10, Yes], [8, 3, 11, 11, Yes], [8, 4, 12, 12, Yes], [8, 5, 0, 0, Yes], [8, 6, 1, 1, Yes], [8, 7, 2, 2, Yes], [8, 8, 3, 3, Yes], [8, 9, 4, 4, Yes], [8, 10, 5, 5, Yes], [8, 11, 6, 6, Yes], [8, 12, 7, 7, Yes], [9, 0, 9, 9, Yes], [9, 1, 10, 10, Yes], [9, 2, 11, 11, Yes], [9, 3, 12, 12, Yes], [9, 4, 0, 0, Yes], [9, 5, 1, 1, Yes], [9, 6, 2, 2, Yes], [9, 7, 3, 3, Yes], [9, 8, 4, 4, Yes], [9, 9, 5, 5, Yes], [9, 10, 6, 6, Yes], [9, 11, 7, 7, Yes], [9, 12, 8, 8, Yes], [10, 0, 10, 10, Yes], [10, 1, 11, 11, Yes], [10, 2, 12, 12, Yes], [10, 3, 0, 0, Yes], [10, 4, 1, 1, Yes], [10, 5, 2, 2, Yes], [10, 6, 3, 3, Yes], [10, 7, 4, 4, Yes], [10, 8, 5, 5, Yes], [10, 9, 6, 6, Yes], [10, 10, 7, 7, Yes], [10, 11, 8, 8, Yes], [10, 12, 9, 9, Yes], [11, 0, 11, 11, Yes], [11, 1, 12, 12, Yes], [11, 2, 0, 0, Yes], [11, 3, 1, 1, Yes], [11, 4, 2, 2, Yes], [11, 5, 3, 3, Yes], [11, 6, 4, 4, Yes], [11, 7, 5, 5, Yes], [11, 8, 6, 6, Yes], [11, 9, 7, 7, Yes], [11, 10, 8, 8, Yes], [11, 11, 9, 9, Yes], [11, 12, 10, 10, Yes], [12, 0, 12, 12, Yes], [12, 1, 0, 0, Yes], [12, 2, 1, 1, Yes], [12, 3, 2, 2, Yes], [12, 4, 3, 3, Yes], [12, 5, 4, 4, Yes], [12, 6, 5, 5, Yes], [12, 7, 6, 6, Yes], [12, 8, 7, 7, Yes], [12, 9, 8, 8, Yes], [12, 10, 9, 9, Yes], [12, 11, 10, 10, Yes], [12, 12, 11, 11, Yes]], Yes, null]

Conditions: none

For these same source and target groups,check f(xy)=f(x)f(y) for every pair and retain the first failed row/null.A total map need not be a homomorphism.Failure blocks kernel theorems;an internal fallback is not the true identity preimage.

Step identifiers and dependencies

Rule: Kernel_complete_TOTAL_map_ALL_product_pairs_before_theorems · Uses: G001, S007

S009 · implies

Before

source roles = [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], 0, 0]

After

derived = [[0], 0, 0]

Conditions: none

The kernel is the complete preimage of the actual target identity.Cancel f(e)=f(e)^2 in the target group to obtain f(e)=e_target,so the true kernel is nonempty;do not assume identity0.

Step identifiers and dependencies

Rule: Kernel_actual_TARGET_identity_complete_preimage · Uses: G001, S008

S010 · implies

Before

source roles = [Kernel ACTUAL identity preimage, [0], 0]

After

derived = [Yes, [[0, 0, 0, Yes]], Yes, [[0, 0, Yes]], Yes, Yes]

Conditions: none

H is nonempty and unique.Check identity membership,all products and every inverse separately;their conjunction qualifies a subgroup.Size divisibility or finiteness alone is insufficient.

Step identifiers and dependencies

Rule: Subgroup_identity_ALL_products_ALL_inverses_separate · Uses: G001, S009

S011 · implies

Before

source roles = [Kernel ACTUAL identity preimage, 0, [0], 0]

After

derived = [[[0, 0, 0, 0]], [0], Yes]

Conditions: none

In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.

Step identifiers and dependencies

Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S010

S012 · implies

Before

source roles = [Kernel ACTUAL identity preimage, 1, [0], 12]

After

derived = [[[0, 1, 12, 0]], [0], Yes]

Conditions: none

In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.

Step identifiers and dependencies

Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S011

S013 · implies

Before

source roles = [Kernel ACTUAL identity preimage, 2, [0], 11]

After

derived = [[[0, 2, 11, 0]], [0], Yes]

Conditions: none

In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.

Step identifiers and dependencies

Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S012

S014 · implies

Before

source roles = [Kernel ACTUAL identity preimage, 3, [0], 10]

After

derived = [[[0, 3, 10, 0]], [0], Yes]

Conditions: none

In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.

Step identifiers and dependencies

Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S013

S015 · implies

Before

source roles = [Kernel ACTUAL identity preimage, 4, [0], 9]

After

derived = [[[0, 4, 9, 0]], [0], Yes]

Conditions: none

In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.

Step identifiers and dependencies

Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S014

S016 · implies

Before

source roles = [Kernel ACTUAL identity preimage, 5, [0], 8]

After

derived = [[[0, 5, 8, 0]], [0], Yes]

Conditions: none

In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.

Step identifiers and dependencies

Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S015

S017 · implies

Before

source roles = [Kernel ACTUAL identity preimage, 6, [0], 7]

After

derived = [[[0, 6, 7, 0]], [0], Yes]

Conditions: none

In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.

Step identifiers and dependencies

Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S016

S018 · implies

Before

source roles = [Kernel ACTUAL identity preimage, 7, [0], 6]

After

derived = [[[0, 7, 6, 0]], [0], Yes]

Conditions: none

In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.

Step identifiers and dependencies

Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S017

S019 · implies

Before

source roles = [Kernel ACTUAL identity preimage, 8, [0], 5]

After

derived = [[[0, 8, 5, 0]], [0], Yes]

Conditions: none

In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.

Step identifiers and dependencies

Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S018

S020 · implies

Before

source roles = [Kernel ACTUAL identity preimage, 9, [0], 4]

After

derived = [[[0, 9, 4, 0]], [0], Yes]

Conditions: none

In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.

Step identifiers and dependencies

Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S019

S021 · implies

Before

source roles = [Kernel ACTUAL identity preimage, 10, [0], 3]

After

derived = [[[0, 10, 3, 0]], [0], Yes]

Conditions: none

In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.

Step identifiers and dependencies

Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S020

S022 · implies

Before

source roles = [Kernel ACTUAL identity preimage, 11, [0], 2]

After

derived = [[[0, 11, 2, 0]], [0], Yes]

Conditions: none

In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.

Step identifiers and dependencies

Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S021

S023 · implies

Before

source roles = [Kernel ACTUAL identity preimage, 12, [0], 1]

After

derived = [[[0, 12, 1, 0]], [0], Yes]

Conditions: none

In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.

Step identifiers and dependencies

Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S022

S024 · implies

Before

source roles = [[0], Yes, 0]

After

derived = [f(kl)=f(k)f(l)=e, f(k^-1)=f(k)^-1=e, f(gkg^-1)=f(g)e*f(g)^-1=e]

Conditions: none

Products,inverses and every conjugate map to target identity,giving the general closure/normality argument alongside this complete finite audit.No quotient/isomorphism is claimed.

Step identifiers and dependencies

Rule: Kernel_GENERAL_closure_normality_from_homomorphism · Uses: G001, S023

S025 · implies

Before

source roles = [0, [0], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]]

After

derived = [[[0, [0]], [1, [1]], [2, [2]], [3, [3]], [4, [4]], [5, [5]], [6, [6]], [7, [7]], [8, [8]], [9, [9]], [10, [10]], [11, [11]], [12, [12]]], Yes, Yes, Yes, No]

Conditions: none

If f is injective,f(k)=f(e) forces k=e.If K={e},image equality gives a^-1b=e and a=b.The fixed No withholds quotient/first-isomorphism claims,not this iff or injectivity.

Step identifiers and dependencies

Rule: Kernel_trivial_actual_identity_iff_injective_NO_quotient_claim · Uses: G001, S024

Reverse checks

  • S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([0], Yes, Yes, Yes, No, trivial_kernel_iff_injective_without_quotient_or_first_isomorphism_claim)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: b21f906409e548c4a59f21f08b360f873a37786d609da1666e1dce36eb20a507
  • candidate record SHA-256: 76b5b4ca6bafd0a1d7139e8d7192ee8b409f7bf8227dc7843c03ce7a19af610f
  • semantic review receipt SHA-256: 049a9e07e69fead88d93b34f0937b9470a06d71c1a8d097058fc83554529bdd0
  • question-first witness SHA-256: 0c71ebb418f3de347d232d159fc25e93036f4ab9166bcfb5ee5b4be86d5c55b9
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON