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R0049-25328 · q-20261015-advance-20261114-r02-finite-quotient-group-0078
Use complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 0, 6, 7, 8, 9, 5], [2, 3, 4, 0, 1, 7, 8, 9, 5, 6], [3, 4, 0, 1, 2, 8, 9, 5, 6, 7], [4, 0, 1, 2, 3, 9, 5, 6, 7, 8], [5, 9, 8, 7, 6, 0, 4, 3, 2, 1], [6, 5, 9, 8, 7, 1, 0, 4, 3, 2], [7, 6, 5, 9, 8, 2, 1, 0, 4, 3], [8, 7, 6, 5, 9, 3, 2, 1, 0, 4], [9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0, explicit subset=[0, 5] and the entire left-coset partition. Subgroup qualification precedes normality. Quotient elements are coset sets, not representatives; coset indices follow the source canonical element-list order. Give (subgroup, normal, cosets, representative_counterexample, well_defined, quotient_group_certificate, boundary).
G001 — Use complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 0, 6, 7, 8, 9, 5], [2, 3, 4, 0, 1, 7, 8, 9, 5, 6], [3, 4, 0, 1, 2, 8, 9, 5, 6, 7], [4, 0, 1, 2, 3, 9, 5, 6, 7, 8], [5, 9, 8, 7, 6, 0, 4, 3, 2, 1], [6, 5, 9, 8, 7, 1, 0, 4, 3, 2], [7, 6, 5, 9, 8, 2, 1, 0, 4, 3], [8, 7, 6, 5, 9, 3, 2, 1, 0, 4], [9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], identity=0, explicit subset=[0, 5] and the entire left-coset partition. Subgroup qualification precedes normality. Quotient elements are coset sets, not representatives; coset indices follow the source canonical element-list order. Give (subgroup, normal, cosets, representative_counterexample, well_defined, quotient_group_certificate, boundary).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 = [quotient-nonnormal-boundary]
After
derived = [[subgroup, normal, cosets, representative_counterexample, well_defined, quotient_group_certificate, boundary]]
Conditions: none
List the full source field order without JavaScript.LP boundary modes all have6 fields but select different roles from the original mode,not answer shape.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
source roles = [[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [1, 2, 3, 4, 0, 6, 7, 8, 9, 5], [2, 3, 4, 0, 1, 7, 8, 9, 5, 6], [3, 4, 0, 1, 2, 8, 9, 5, 6, 7], [4, 0, 1, 2, 3, 9, 5, 6, 7, 8], [5, 9, 8, 7, 6, 0, 4, 3, 2, 1], [6, 5, 9, 8, 7, 1, 0, 4, 3, 2], [7, 6, 5, 9, 8, 2, 1, 0, 4, 3], [8, 7, 6, 5, 9, 3, 2, 1, 0, 4], [9, 8, 7, 6, 5, 4, 3, 2, 1, 0]], 0]
After
derived = [10, [[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]], Yes]
Conditions: none
Validate the complete closed Cayley table and actual identity label,without assuming0 or the old32-element cap.
Rule: Quotient_complete_group_table_actual_identity · Uses: G001, S001
Before
source roles = [10, 0]
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]], 1000, [0, 4, 3, 2, 1, 5, 6, 7, 8, 9]]
Conditions: none
Each pair-matrix entry checks associativity for every z,covering n^3 obligations;each inverse returns the same identity from both sides.This is exhaustive,not sampling.
Rule: Quotient_ALL_group_associativity_triples_and_two_sided_inverses · Uses: G001, S002
Before
source roles = [0, [0, 5]]
After
derived = [[[0, 0, 0, Yes], [0, 5, 5, Yes], [5, 0, 5, Yes], [5, 5, 0, Yes]], [[0, 0, Yes], [5, 5, Yes]], Yes, [[0, [0, 5]], [1, [0, 7]], [2, [0, 9]], [3, [0, 6]], [4, [0, 8]], [5, [0, 5]], [6, [0, 7]], [7, [0, 9]], [8, [0, 6]], [9, [0, 8]]], No]
Conditions: none
First qualify identity,all products and inverse closure as a genuine subgroup,then test conjugation for every g.Positive branches require normality and the boundary requires actual nonnormality,not failed subgroup qualification.
Rule: Quotient_genuine_subgroup_before_actual_normality · Uses: G001, S003
Before
source roles = [[0, 5], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]]
After
derived = [[[0, 5], [1, 6], [2, 7], [3, 8], [4, 9]], [0, 1, 2, 3, 4, 0, 1, 2, 3, 4], Yes]
Conditions: none
Every g belongs to gH.If aH meets bH,ah1=bh2 implies b^-1a in H and the complete cosets coincide.Enumerate every g,deduplicate and retain source order.Quotient elements are coset SETS,not least representatives.
Rule: Quotient_ALL_LEFT_coset_SETS_complete_disjoint_partition · Uses: G001, S004
Before
source roles = [0, 0, [0, 5], [0, 5]]
After
derived = [[[0, 0, 0, 0], [0, 5, 5, 0], [5, 0, 5, 0], [5, 5, 0, 0]], [0], Yes]
Conditions: none
For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices.
Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S005
Before
source roles = [0, 1, [0, 5], [1, 6]]
After
derived = [[[0, 1, 1, 1], [0, 6, 6, 1], [5, 1, 9, 4], [5, 6, 4, 4]], [1, 4], No]
Conditions: none
For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices.
Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S006
Before
source roles = [0, 2, [0, 5], [2, 7]]
After
derived = [[[0, 2, 2, 2], [0, 7, 7, 2], [5, 2, 8, 3], [5, 7, 3, 3]], [2, 3], No]
Conditions: none
For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices.
Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S007
Before
source roles = [0, 3, [0, 5], [3, 8]]
After
derived = [[[0, 3, 3, 3], [0, 8, 8, 3], [5, 3, 7, 2], [5, 8, 2, 2]], [2, 3], No]
Conditions: none
For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices.
Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S008
Before
source roles = [0, 4, [0, 5], [4, 9]]
After
derived = [[[0, 4, 4, 4], [0, 9, 9, 4], [5, 4, 6, 1], [5, 9, 1, 1]], [1, 4], No]
Conditions: none
For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices.
Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S009
Before
source roles = [1, 0, [1, 6], [0, 5]]
After
derived = [[[1, 0, 1, 1], [1, 5, 6, 1], [6, 0, 6, 1], [6, 5, 1, 1]], [1], Yes]
Conditions: none
For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices.
Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S010
Before
source roles = [1, 1, [1, 6], [1, 6]]
After
derived = [[[1, 1, 2, 2], [1, 6, 7, 2], [6, 1, 5, 0], [6, 6, 0, 0]], [0, 2], No]
Conditions: none
For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices.
Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S011
Before
source roles = [1, 2, [1, 6], [2, 7]]
After
derived = [[[1, 2, 3, 3], [1, 7, 8, 3], [6, 2, 9, 4], [6, 7, 4, 4]], [3, 4], No]
Conditions: none
For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices.
Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S012
Before
source roles = [1, 3, [1, 6], [3, 8]]
After
derived = [[[1, 3, 4, 4], [1, 8, 9, 4], [6, 3, 8, 3], [6, 8, 3, 3]], [3, 4], No]
Conditions: none
For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices.
Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S013
Before
source roles = [1, 4, [1, 6], [4, 9]]
After
derived = [[[1, 4, 0, 0], [1, 9, 5, 0], [6, 4, 7, 2], [6, 9, 2, 2]], [0, 2], No]
Conditions: none
For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices.
Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S014
Before
source roles = [2, 0, [2, 7], [0, 5]]
After
derived = [[[2, 0, 2, 2], [2, 5, 7, 2], [7, 0, 7, 2], [7, 5, 2, 2]], [2], Yes]
Conditions: none
For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices.
Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S015
Before
source roles = [2, 1, [2, 7], [1, 6]]
After
derived = [[[2, 1, 3, 3], [2, 6, 8, 3], [7, 1, 6, 1], [7, 6, 1, 1]], [1, 3], No]
Conditions: none
For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices.
Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S016
Before
source roles = [2, 2, [2, 7], [2, 7]]
After
derived = [[[2, 2, 4, 4], [2, 7, 9, 4], [7, 2, 5, 0], [7, 7, 0, 0]], [0, 4], No]
Conditions: none
For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices.
Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S017
Before
source roles = [2, 3, [2, 7], [3, 8]]
After
derived = [[[2, 3, 0, 0], [2, 8, 5, 0], [7, 3, 9, 4], [7, 8, 4, 4]], [0, 4], No]
Conditions: none
For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices.
Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S018
Before
source roles = [2, 4, [2, 7], [4, 9]]
After
derived = [[[2, 4, 1, 1], [2, 9, 6, 1], [7, 4, 8, 3], [7, 9, 3, 3]], [1, 3], No]
Conditions: none
For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices.
Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S019
Before
source roles = [3, 0, [3, 8], [0, 5]]
After
derived = [[[3, 0, 3, 3], [3, 5, 8, 3], [8, 0, 8, 3], [8, 5, 3, 3]], [3], Yes]
Conditions: none
For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices.
Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S020
Before
source roles = [3, 1, [3, 8], [1, 6]]
After
derived = [[[3, 1, 4, 4], [3, 6, 9, 4], [8, 1, 7, 2], [8, 6, 2, 2]], [2, 4], No]
Conditions: none
For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices.
Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S021
Before
source roles = [3, 2, [3, 8], [2, 7]]
After
derived = [[[3, 2, 0, 0], [3, 7, 5, 0], [8, 2, 6, 1], [8, 7, 1, 1]], [0, 1], No]
Conditions: none
For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices.
Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S022
Before
source roles = [3, 3, [3, 8], [3, 8]]
After
derived = [[[3, 3, 1, 1], [3, 8, 6, 1], [8, 3, 5, 0], [8, 8, 0, 0]], [0, 1], No]
Conditions: none
For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices.
Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S023
Before
source roles = [3, 4, [3, 8], [4, 9]]
After
derived = [[[3, 4, 2, 2], [3, 9, 7, 2], [8, 4, 9, 4], [8, 9, 4, 4]], [2, 4], No]
Conditions: none
For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices.
Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S024
Before
source roles = [4, 0, [4, 9], [0, 5]]
After
derived = [[[4, 0, 4, 4], [4, 5, 9, 4], [9, 0, 9, 4], [9, 5, 4, 4]], [4], Yes]
Conditions: none
For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices.
Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S025
Before
source roles = [4, 1, [4, 9], [1, 6]]
After
derived = [[[4, 1, 0, 0], [4, 6, 5, 0], [9, 1, 8, 3], [9, 6, 3, 3]], [0, 3], No]
Conditions: none
For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices.
Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S026
Before
source roles = [4, 2, [4, 9], [2, 7]]
After
derived = [[[4, 2, 1, 1], [4, 7, 6, 1], [9, 2, 7, 2], [9, 7, 2, 2]], [1, 2], No]
Conditions: none
For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices.
Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S027
Before
source roles = [4, 3, [4, 9], [3, 8]]
After
derived = [[[4, 3, 2, 2], [4, 8, 7, 2], [9, 3, 6, 1], [9, 8, 1, 1]], [1, 2], No]
Conditions: none
For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices.
Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S028
Before
source roles = [4, 4, [4, 9], [4, 9]]
After
derived = [[[4, 4, 3, 3], [4, 9, 8, 3], [9, 4, 5, 0], [9, 9, 0, 0]], [0, 3], No]
Conditions: none
For every ordered coset pair,all representative products must land in one product coset.A least-representative table alone does not establish well-definedness;retain the complete set of possible indices.
Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S029
Before
source roles = [[0, 1, [1, 4], No], [0, 5], [1, 6]]
After
derived = [[0, 1, 0, 1, 1, 5, 1, 4], No, No]
Conditions: none
Keep the source first bad ordered pair and left,left2,right,right2 search order.Different representatives of the same two cosets give different product cosets,so multiplication fails to be well-defined.The subgroup and coset partition remain valid;do not invent a quotient table.
Rule: Quotient_FIRST_bad_pair_exact_source_nested_representative_witness · Uses: G001, S030
S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.([0, 5], No, [[0, 5], [1, 6], [2, 7], [3, 8], [4, 9]], [0, 1, 0, 1, 1, 5, 1, 4], No, No, explicit_representative_dependence_blocks_quotient_group_multiplication)
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: ddb447212d8fa1b55862dbfee5f7276880bfd2b519ca38cde35fc14976772202candidate record SHA-256: 8691a03837c60c89c75ac98a56ff01d5f557d0851a81ac58d35f342b45da3898semantic review receipt SHA-256: 6212b8b0b3b273f3a514e26ebd17d7e81219c56a8f47e40047ee2b6bf01d1ba3question-first witness SHA-256: f32a23d5e1b66c7794c60def58741bcdfacff24f3cdfc9bc48109c8f71543cf3Lean: NOT_RUN · human review: NOT_PERFORMED