Question 25378: Every Representative Pair Must Land in the Same Declared Product Coset

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Original question and givens
Definition and scope
Step-by-step derivation
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Result
This question review and source binding

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R0049-25378 · q-20261015-advance-20261114-r02-finite-quotient-group-0053

Original question and givens

Use complete finite group table=[[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], [5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6]], identity=0, explicit subset=[0, 2, 4, 6] 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, representative_rows, well_defined, certificate).

G001 — Use complete finite group table=[[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], [5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6]], identity=0, explicit subset=[0, 2, 4, 6] 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, representative_rows, well_defined, 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-quotient-group

Step-by-step derivation

S001 · implies

Before

source roles = [quotient-representative-well-definedness]

After

derived = [[cosets, representative_rows, well_defined, 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.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · implies

Before

source roles = [[[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], [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]

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 Cayley table and actual identity label,without assuming0 or the old32-element cap.

Step identifiers and dependencies

Rule: Quotient_complete_group_table_actual_identity · Uses: G001, S001

S003 · implies

Before

source roles = [8, 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]], 512, [0, 7, 6, 5, 4, 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.

Step identifiers and dependencies

Rule: Quotient_ALL_group_associativity_triples_and_two_sided_inverses · Uses: G001, S002

S004 · implies

Before

source roles = [0, [0, 2, 4, 6]]

After

derived = [[[0, 0, 0, Yes], [0, 2, 2, Yes], [0, 4, 4, Yes], [0, 6, 6, Yes], [2, 0, 2, Yes], [2, 2, 4, Yes], [2, 4, 6, Yes], [2, 6, 0, Yes], [4, 0, 4, Yes], [4, 2, 6, Yes], [4, 4, 0, Yes], [4, 6, 2, Yes], [6, 0, 6, Yes], [6, 2, 0, Yes], [6, 4, 2, Yes], [6, 6, 4, Yes]], [[0, 0, Yes], [2, 6, Yes], [4, 4, Yes], [6, 2, Yes]], Yes, [[0, [0, 2, 4, 6]], [1, [0, 2, 4, 6]], [2, [0, 2, 4, 6]], [3, [0, 2, 4, 6]], [4, [0, 2, 4, 6]], [5, [0, 2, 4, 6]], [6, [0, 2, 4, 6]], [7, [0, 2, 4, 6]]], 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.

Step identifiers and dependencies

Rule: Quotient_genuine_subgroup_before_actual_normality · Uses: G001, S003

S005 · implies

Before

source roles = [[0, 2, 4, 6], [0, 1, 2, 3, 4, 5, 6, 7]]

After

derived = [[[0, 2, 4, 6], [1, 3, 5, 7]], [0, 1, 0, 1, 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.

Step identifiers and dependencies

Rule: Quotient_ALL_LEFT_coset_SETS_complete_disjoint_partition · Uses: G001, S004

S006 · implies

Before

source roles = [0, 0, [0, 2, 4, 6], [0, 2, 4, 6]]

After

derived = [[[0, 0, 0, 0], [0, 2, 2, 0], [0, 4, 4, 0], [0, 6, 6, 0], [2, 0, 2, 0], [2, 2, 4, 0], [2, 4, 6, 0], [2, 6, 0, 0], [4, 0, 4, 0], [4, 2, 6, 0], [4, 4, 0, 0], [4, 6, 2, 0], [6, 0, 6, 0], [6, 2, 0, 0], [6, 4, 2, 0], [6, 6, 4, 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.

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S005

S007 · implies

Before

source roles = [0, 1, [0, 2, 4, 6], [1, 3, 5, 7]]

After

derived = [[[0, 1, 1, 1], [0, 3, 3, 1], [0, 5, 5, 1], [0, 7, 7, 1], [2, 1, 3, 1], [2, 3, 5, 1], [2, 5, 7, 1], [2, 7, 1, 1], [4, 1, 5, 1], [4, 3, 7, 1], [4, 5, 1, 1], [4, 7, 3, 1], [6, 1, 7, 1], [6, 3, 1, 1], [6, 5, 3, 1], [6, 7, 5, 1]], [1], Yes]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S006

S008 · implies

Before

source roles = [1, 0, [1, 3, 5, 7], [0, 2, 4, 6]]

After

derived = [[[1, 0, 1, 1], [1, 2, 3, 1], [1, 4, 5, 1], [1, 6, 7, 1], [3, 0, 3, 1], [3, 2, 5, 1], [3, 4, 7, 1], [3, 6, 1, 1], [5, 0, 5, 1], [5, 2, 7, 1], [5, 4, 1, 1], [5, 6, 3, 1], [7, 0, 7, 1], [7, 2, 1, 1], [7, 4, 3, 1], [7, 6, 5, 1]], [1], Yes]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S007

S009 · implies

Before

source roles = [1, 1, [1, 3, 5, 7], [1, 3, 5, 7]]

After

derived = [[[1, 1, 2, 0], [1, 3, 4, 0], [1, 5, 6, 0], [1, 7, 0, 0], [3, 1, 4, 0], [3, 3, 6, 0], [3, 5, 0, 0], [3, 7, 2, 0], [5, 1, 6, 0], [5, 3, 0, 0], [5, 5, 2, 0], [5, 7, 4, 0], [7, 1, 0, 0], [7, 3, 2, 0], [7, 5, 4, 0], [7, 7, 6, 0]], [0], Yes]

Conditions: none

Step identifiers and dependencies

Rule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S008

S010 · implies

Before

source roles = [[0, 2, 4, 6], Yes, [[0, 2, 4, 6], [1, 3, 5, 7]]]

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.

Step identifiers and dependencies

Rule: Quotient_GENERAL_normality_representative_replacement_and_group_laws · Uses: G001, S009

Reverse checks

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, 2, 4, 6], [1, 3, 5, 7]], [[0, 0, [0], Yes], [0, 1, [1], Yes], [1, 0, [1], Yes], [1, 1, [0], Yes]], Yes, every_representative_pair_lands_in_one_declared_product_coset)

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: c6984c2448b6850412eee58dbd52871621bb3affa80c2316e5f4e98e16db193d
candidate record SHA-256: 4282c61022efba90f3ee8828c8f30b54122e3368d1a97b36833e5ceffb04b512
semantic review receipt SHA-256: 6212b8b0b3b273f3a514e26ebd17d7e81219c56a8f47e40047ee2b6bf01d1ba3
question-first witness SHA-256: f32a23d5e1b66c7794c60def58741bcdfacff24f3cdfc9bc48109c8f71543cf3
Lean: NOT_RUN · human review: NOT_PERFORMED

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