# Question 24927: Qualify Both Groups and Every Product Pair before Applying Kernel Theorems

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0048-24927 · q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0002



## Original question and givens

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).


- G001 — 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).





## 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.


- [mko-finite-group-homomorphism-kernel](https://ocme.evemisslab.com/en/mko/mko-finite-group-homomorphism-kernel/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [kernel-homomorphism-qualification]

After

derived = [[source_order, target_order, mapping, product_law_rows, homomorphism, first_failed_pair_or_null, certificate]]

Conditions: none

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

Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001





S002 · impliesBefore

source roles = [Kernel ORIGINAL SOURCE group, 64, [[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 0]

After

derived = [4, [[0, 0, 0, 0], [1, 1, 1, 1], [2, 2, 2, 2], [3, 3, 3, 3]], 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 dependenciesRule: Group_complete_Cayley_signature_actual_identity · Uses: G001, S001





S003 · impliesBefore

source roles = [Kernel ORIGINAL SOURCE group, 4, 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]], 64, 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 dependenciesRule: Group_ALL_associativity_triples_compact_complete_pair_matrix · Uses: G001, S002





S004 · impliesBefore

source roles = [Kernel ORIGINAL SOURCE group, 0]

After

derived = [[0, 3, 2, 1], [[0, 0, 0, 0], [1, 3, 0, 0], [2, 2, 0, 0], [3, 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 dependenciesRule: Group_ALL_two_sided_inverses · Uses: G001, S003





S005 · impliesBefore

source roles = [Kernel ORIGINAL TARGET group, 64, [[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], 0]

After

derived = [4, [[0, 0, 0, 0], [1, 1, 1, 1], [2, 2, 2, 2], [3, 3, 3, 3]], 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 dependenciesRule: Group_complete_Cayley_signature_actual_identity · Uses: G001, S004





S006 · impliesBefore

source roles = [Kernel ORIGINAL TARGET group, 4, 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]], 64, 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 dependenciesRule: Group_ALL_associativity_triples_compact_complete_pair_matrix · Uses: G001, S005





S007 · impliesBefore

source roles = [Kernel ORIGINAL TARGET group, 0]

After

derived = [[0, 3, 2, 1], [[0, 0, 0, 0], [1, 3, 0, 0], [2, 2, 0, 0], [3, 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 dependenciesRule: Group_ALL_two_sided_inverses · Uses: G001, S006





S008 · impliesBefore

source roles = [[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], [[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], [0, 3, 2, 1]]

After

derived = [[[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]

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 dependenciesRule: Kernel_complete_TOTAL_map_ALL_product_pairs_before_theorems · Uses: G001, S007









## Reverse checks


- 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

(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)

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: 3473c69c9d4c58e65487956c23036e6b098ac71ce3d8a0964207d630909fe487
- candidate record SHA-256: 04e0fed177af10c298e4a6e071c2fa15fe49c00bbd3973801715a2dae92bccfd
- semantic review receipt SHA-256: 049a9e07e69fead88d93b34f0937b9470a06d71c1a8d097058fc83554529bdd0
- question-first witness SHA-256: 0c71ebb418f3de347d232d159fc25e93036f4ab9166bcfb5ee5b4be86d5c55b9
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0002.json)