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R0049-25301 · q-20261015-advance-20261114-r02-finite-quotient-group-0026
Use complete finite group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, explicit subset=[0, 2] 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 (cosets, multiplication_table, identity_coset, inverse_rows, well_defined, commutativity_certificate, certificate).
G001 — Use complete finite group table=[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 0, 1, 2]], identity=0, explicit subset=[0, 2] 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 (cosets, multiplication_table, identity_coset, inverse_rows, well_defined, commutativity_certificate, 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 = [quotient-coset-multiplication-table]
After
derived = [[cosets, multiplication_table, identity_coset, inverse_rows, well_defined, commutativity_certificate, certificate]]
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], [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 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 = [4, 0]
After
derived = [[[Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes]], 64, [0, 3, 2, 1]]
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, 2]]
After
derived = [[[0, 0, 0, Yes], [0, 2, 2, Yes], [2, 0, 2, Yes], [2, 2, 0, Yes]], [[0, 0, Yes], [2, 2, Yes]], Yes, [[0, [0, 2]], [1, [0, 2]], [2, [0, 2]], [3, [0, 2]]], Yes]
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, 2], [0, 1, 2, 3]]
After
derived = [[[0, 2], [1, 3]], [0, 1, 0, 1], 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, 2], [0, 2]]
After
derived = [[[0, 0, 0, 0], [0, 2, 2, 0], [2, 0, 2, 0], [2, 2, 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, 2], [1, 3]]
After
derived = [[[0, 1, 1, 1], [0, 3, 3, 1], [2, 1, 3, 1], [2, 3, 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, S006
Before
source roles = [1, 0, [1, 3], [0, 2]]
After
derived = [[[1, 0, 1, 1], [1, 2, 3, 1], [3, 0, 3, 1], [3, 2, 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, S007
Before
source roles = [1, 1, [1, 3], [1, 3]]
After
derived = [[[1, 1, 2, 0], [1, 3, 0, 0], [3, 1, 0, 0], [3, 3, 2, 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, S008
Before
source roles = [[0, 2], Yes, [[0, 2], [1, 3]]]
After
derived = [[[0, 1], [1, 0]], 0, [[0, 0], [1, 1]], Yes, No]
Conditions: none
Replacing a by ah and b by bk yields(ah)(bk)=ab(b^-1hb)k in abH,using normality for representative independence.Associativity descends from G,eH is identity and a^-1H inverse.The fixed No withholds automatic commutativity;it does not call a particular cyclic quotient noncommutative or invoke first isomorphism.
Rule: Quotient_GENERAL_normality_representative_replacement_and_group_laws · Uses: G001, S009
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, 2], [1, 3]], [[0, 1], [1, 0]], 0, [[0, 0], [1, 1]], Yes, No, coset_product_preserves_identity_and_inverse_but_need_not_be_commutative)
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: 73c792413da8af5eaf68da1e378a38af4a4bce2c08d9dc9f1ad951a97d0dcc5ecandidate record SHA-256: 9835c6a7aa72111b93bd8f0291c5394438db30bb098635c9abb13b33fa8b05ffsemantic review receipt SHA-256: 6212b8b0b3b273f3a514e26ebd17d7e81219c56a8f47e40047ee2b6bf01d1ba3question-first witness SHA-256: f32a23d5e1b66c7794c60def58741bcdfacff24f3cdfc9bc48109c8f71543cf3Lean: NOT_RUN · human review: NOT_PERFORMED