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Question 24587: Subgroup Closure Precedes a Normal-Subgroup Certificate

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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R0048-24587 · q-20261015-advance-20261113-r01-finite-normal-subgroup-0012

Original question and givens

Use complete finite Cayley table=[[4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4], [0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2]], identity=2 and candidate subset H=[0, 1, 2, 3, 4, 5]. Row g/column h gives g*h. Validate all group axioms, then identity/product/inverse closure of the whole subset before treating it as a subgroup. Normality has a separate all-g obligation. Give (subset, contains_identity, product_closed, inverse_closed, subgroup, normal_subgroup, certificate).

  • G001 — Use complete finite Cayley table=[[4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4], [0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2]], identity=2 and candidate subset H=[0, 1, 2, 3, 4, 5]. Row g/column h gives g*h. Validate all group axioms, then identity/product/inverse closure of the whole subset before treating it as a subgroup. Normality has a separate all-g obligation. Give (subset, contains_identity, product_closed, inverse_closed, subgroup, normal_subgroup, 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.

Step-by-step derivation

S001 · implies

Before

source roles = [normal-subgroup-qualification]

After

derived = [[subset, contains_identity, product_closed, inverse_closed, subgroup, normal_subgroup, certificate]]

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 = [Normality ORIGINAL group, 32, [[4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4], [0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2]], 2]

After

derived = [6, [[0, 0, 0, 0], [1, 1, 1, 1], [2, 2, 2, 2], [3, 3, 3, 3], [4, 4, 4, 4], [5, 5, 5, 5]], 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 = [Normality ORIGINAL group, 6, 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]], 216, 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 = [Normality ORIGINAL group, 2]

After

derived = [[4, 3, 2, 1, 0, 5], [[0, 4, 2, 2], [1, 3, 2, 2], [2, 2, 2, 2], [3, 1, 2, 2], [4, 0, 2, 2], [5, 5, 2, 2]], 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 = [Normality ORIGINAL H, [0, 1, 2, 3, 4, 5], 2]

After

derived = [Yes, [[0, 0, 4, Yes], [0, 1, 5, Yes], [0, 2, 0, Yes], [0, 3, 1, Yes], [0, 4, 2, Yes], [0, 5, 3, Yes], [1, 0, 5, Yes], [1, 1, 0, Yes], [1, 2, 1, Yes], [1, 3, 2, Yes], [1, 4, 3, Yes], [1, 5, 4, Yes], [2, 0, 0, Yes], [2, 1, 1, Yes], [2, 2, 2, Yes], [2, 3, 3, Yes], [2, 4, 4, Yes], [2, 5, 5, Yes], [3, 0, 1, Yes], [3, 1, 2, Yes], [3, 2, 3, Yes], [3, 3, 4, Yes], [3, 4, 5, Yes], [3, 5, 0, Yes], [4, 0, 2, Yes], [4, 1, 3, Yes], [4, 2, 4, Yes], [4, 3, 5, Yes], [4, 4, 0, Yes], [4, 5, 1, Yes], [5, 0, 3, Yes], [5, 1, 4, Yes], [5, 2, 5, Yes], [5, 3, 0, Yes], [5, 4, 1, Yes], [5, 5, 2, Yes]], Yes, [[0, 4, Yes], [1, 3, Yes], [2, 2, Yes], [3, 1, Yes], [4, 0, Yes], [5, 5, 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, S004

S006 · implies

Before

source roles = [Normality ORIGINAL H, 0, [0, 1, 2, 3, 4, 5], 4]

After

derived = [[[0, 4, 4, 0], [1, 5, 4, 1], [2, 0, 4, 2], [3, 1, 4, 3], [4, 2, 4, 4], [5, 3, 4, 5]], [0, 1, 2, 3, 4, 5], 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, S005

S007 · implies

Before

source roles = [Normality ORIGINAL H, 1, [0, 1, 2, 3, 4, 5], 3]

After

derived = [[[0, 5, 3, 0], [1, 0, 3, 1], [2, 1, 3, 2], [3, 2, 3, 3], [4, 3, 3, 4], [5, 4, 3, 5]], [0, 1, 2, 3, 4, 5], 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, S006

S008 · implies

Before

source roles = [Normality ORIGINAL H, 2, [0, 1, 2, 3, 4, 5], 2]

After

derived = [[[0, 0, 2, 0], [1, 1, 2, 1], [2, 2, 2, 2], [3, 3, 2, 3], [4, 4, 2, 4], [5, 5, 2, 5]], [0, 1, 2, 3, 4, 5], 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, S007

S009 · implies

Before

source roles = [Normality ORIGINAL H, 3, [0, 1, 2, 3, 4, 5], 1]

After

derived = [[[0, 1, 1, 0], [1, 2, 1, 1], [2, 3, 1, 2], [3, 4, 1, 3], [4, 5, 1, 4], [5, 0, 1, 5]], [0, 1, 2, 3, 4, 5], 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, S008

S010 · implies

Before

source roles = [Normality ORIGINAL H, 4, [0, 1, 2, 3, 4, 5], 0]

After

derived = [[[0, 2, 0, 0], [1, 3, 0, 1], [2, 4, 0, 2], [3, 5, 0, 3], [4, 0, 0, 4], [5, 1, 0, 5]], [0, 1, 2, 3, 4, 5], 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, S009

S011 · implies

Before

source roles = [Normality ORIGINAL H, 5, [0, 1, 2, 3, 4, 5], 5]

After

derived = [[[0, 3, 5, 0], [1, 4, 5, 1], [2, 5, 5, 2], [3, 0, 5, 3], [4, 1, 5, 4], [5, 2, 5, 5]], [0, 1, 2, 3, 4, 5], 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

Reverse checks

  • 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, 1, 2, 3, 4, 5], Yes, Yes, Yes, Yes, Yes, subgroup_obligations_precede_normality)

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: dfbcd5c8641af543548559f18e4307c16a31ae798f2666659104a6ef2eff4d33
  • candidate record SHA-256: dacc2fb682b24f591c72e17f09e08f1e271749fc810d6c45db870e8050b6f00d
  • semantic review receipt SHA-256: 049a9e07e69fead88d93b34f0937b9470a06d71c1a8d097058fc83554529bdd0
  • question-first witness SHA-256: 0c71ebb418f3de347d232d159fc25e93036f4ab9166bcfb5ee5b4be86d5c55b9
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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