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R0048-24593 · q-20261015-advance-20261113-r01-finite-normal-subgroup-0018
Use complete finite Cayley table=[[5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6], [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]], identity=3 and candidate subset H=[1, 3, 5, 7]. 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=[[5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6], [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]], identity=3 and candidate subset H=[1, 3, 5, 7]. 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).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.
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.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
source roles = [Normality ORIGINAL group, 32, [[5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6], [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]], 3]
After
derived = [8, [[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]], 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.
Rule: Group_complete_Cayley_signature_actual_identity · Uses: G001, S001
Before
source roles = [Normality ORIGINAL group, 8, 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]], 512, 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.
Rule: Group_ALL_associativity_triples_compact_complete_pair_matrix · Uses: G001, S002
Before
source roles = [Normality ORIGINAL group, 3]
After
derived = [[6, 5, 4, 3, 2, 1, 0, 7], [[0, 6, 3, 3], [1, 5, 3, 3], [2, 4, 3, 3], [3, 3, 3, 3], [4, 2, 3, 3], [5, 1, 3, 3], [6, 0, 3, 3], [7, 7, 3, 3]], Yes]
Conditions: none
Every left and right inverse must return the same actual identity;associativity gives uniqueness.This completes the group premises.
Rule: Group_ALL_two_sided_inverses · Uses: G001, S003
Before
source roles = [Normality ORIGINAL H, [1, 3, 5, 7], 3]
After
derived = [Yes, [[1, 1, 7, Yes], [1, 3, 1, Yes], [1, 5, 3, Yes], [1, 7, 5, Yes], [3, 1, 1, Yes], [3, 3, 3, Yes], [3, 5, 5, Yes], [3, 7, 7, Yes], [5, 1, 3, Yes], [5, 3, 5, Yes], [5, 5, 7, Yes], [5, 7, 1, Yes], [7, 1, 5, Yes], [7, 3, 7, Yes], [7, 5, 1, Yes], [7, 7, 3, Yes]], Yes, [[1, 5, Yes], [3, 3, Yes], [5, 1, Yes], [7, 7, 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.
Rule: Subgroup_identity_ALL_products_ALL_inverses_separate · Uses: G001, S004
Before
source roles = [Normality ORIGINAL H, 0, [1, 3, 5, 7], 6]
After
derived = [[[1, 6, 6, 1], [3, 0, 6, 3], [5, 2, 6, 5], [7, 4, 6, 7]], [1, 3, 5, 7], 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.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S005
Before
source roles = [Normality ORIGINAL H, 1, [1, 3, 5, 7], 5]
After
derived = [[[1, 7, 5, 1], [3, 1, 5, 3], [5, 3, 5, 5], [7, 5, 5, 7]], [1, 3, 5, 7], 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.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S006
Before
source roles = [Normality ORIGINAL H, 2, [1, 3, 5, 7], 4]
After
derived = [[[1, 0, 4, 1], [3, 2, 4, 3], [5, 4, 4, 5], [7, 6, 4, 7]], [1, 3, 5, 7], 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.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S007
Before
source roles = [Normality ORIGINAL H, 3, [1, 3, 5, 7], 3]
After
derived = [[[1, 1, 3, 1], [3, 3, 3, 3], [5, 5, 3, 5], [7, 7, 3, 7]], [1, 3, 5, 7], 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.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S008
Before
source roles = [Normality ORIGINAL H, 4, [1, 3, 5, 7], 2]
After
derived = [[[1, 2, 2, 1], [3, 4, 2, 3], [5, 6, 2, 5], [7, 0, 2, 7]], [1, 3, 5, 7], 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.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S009
Before
source roles = [Normality ORIGINAL H, 5, [1, 3, 5, 7], 1]
After
derived = [[[1, 3, 1, 1], [3, 5, 1, 3], [5, 7, 1, 5], [7, 1, 1, 7]], [1, 3, 5, 7], 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.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S010
Before
source roles = [Normality ORIGINAL H, 6, [1, 3, 5, 7], 0]
After
derived = [[[1, 4, 0, 1], [3, 6, 0, 3], [5, 0, 0, 5], [7, 2, 0, 7]], [1, 3, 5, 7], 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.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S011
Before
source roles = [Normality ORIGINAL H, 7, [1, 3, 5, 7], 7]
After
derived = [[[1, 5, 7, 1], [3, 7, 7, 3], [5, 1, 7, 5], [7, 3, 7, 7]], [1, 3, 5, 7], 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.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S012
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.([1, 3, 5, 7], 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.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 2cf7031f1ad3442084c2ec7189d65586945f578df7638ce629b993d1d9548872candidate record SHA-256: e9263e3d7166abfa2845889f8ba3b0deed509fe4e33a783440ecbeaa8e3e6126semantic review receipt SHA-256: 049a9e07e69fead88d93b34f0937b9470a06d71c1a8d097058fc83554529bdd0question-first witness SHA-256: 0c71ebb418f3de347d232d159fc25e93036f4ab9166bcfb5ee5b4be86d5c55b9Lean: NOT_RUN · human review: NOT_PERFORMED