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Question 25357: A Quotient Carrier Consists of Complete Coset Sets

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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R0049-25357 · q-20261015-advance-20261114-r02-finite-quotient-group-0007

Original question and givens

Use complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]], identity=0, explicit subset=[0, 2, 4, 6, 8, 10, 12, 14] 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, partition_certificate, coset_count, certificate).

  • G001 — Use complete finite group table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]], identity=0, explicit subset=[0, 2, 4, 6, 8, 10, 12, 14] 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, partition_certificate, coset_count, 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 = [quotient-normal-coset-partition]

After

derived = [[subgroup, normal, cosets, partition_certificate, coset_count, 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, 8, 9, 10, 11, 12, 13, 14, 15], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]], 0]

After

derived = [16, [[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], [10, 10, 10, 10], [11, 11, 11, 11], [12, 12, 12, 12], [13, 13, 13, 13], [14, 14, 14, 14], [15, 15, 15, 15]], 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 = [16, 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, 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, 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]], 4096, [0, 15, 14, 13, 12, 11, 10, 9, 8, 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, 8, 10, 12, 14]]

After

derived = [[[0, 0, 0, Yes], [0, 2, 2, Yes], [0, 4, 4, Yes], [0, 6, 6, Yes], [0, 8, 8, Yes], [0, 10, 10, Yes], [0, 12, 12, Yes], [0, 14, 14, Yes], [2, 0, 2, Yes], [2, 2, 4, Yes], [2, 4, 6, Yes], [2, 6, 8, Yes], [2, 8, 10, Yes], [2, 10, 12, Yes], [2, 12, 14, Yes], [2, 14, 0, Yes], [4, 0, 4, Yes], [4, 2, 6, Yes], [4, 4, 8, Yes], [4, 6, 10, Yes], [4, 8, 12, Yes], [4, 10, 14, Yes], [4, 12, 0, Yes], [4, 14, 2, Yes], [6, 0, 6, Yes], [6, 2, 8, Yes], [6, 4, 10, Yes], [6, 6, 12, Yes], [6, 8, 14, Yes], [6, 10, 0, Yes], [6, 12, 2, Yes], [6, 14, 4, Yes], [8, 0, 8, Yes], [8, 2, 10, Yes], [8, 4, 12, Yes], [8, 6, 14, Yes], [8, 8, 0, Yes], [8, 10, 2, Yes], [8, 12, 4, Yes], [8, 14, 6, Yes], [10, 0, 10, Yes], [10, 2, 12, Yes], [10, 4, 14, Yes], [10, 6, 0, Yes], [10, 8, 2, Yes], [10, 10, 4, Yes], [10, 12, 6, Yes], [10, 14, 8, Yes], [12, 0, 12, Yes], [12, 2, 14, Yes], [12, 4, 0, Yes], [12, 6, 2, Yes], [12, 8, 4, Yes], [12, 10, 6, Yes], [12, 12, 8, Yes], [12, 14, 10, Yes], [14, 0, 14, Yes], [14, 2, 0, Yes], [14, 4, 2, Yes], [14, 6, 4, Yes], [14, 8, 6, Yes], [14, 10, 8, Yes], [14, 12, 10, Yes], [14, 14, 12, Yes]], [[0, 0, Yes], [2, 14, Yes], [4, 12, Yes], [6, 10, Yes], [8, 8, Yes], [10, 6, Yes], [12, 4, Yes], [14, 2, Yes]], Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14]], [1, [0, 2, 4, 6, 8, 10, 12, 14]], [2, [0, 2, 4, 6, 8, 10, 12, 14]], [3, [0, 2, 4, 6, 8, 10, 12, 14]], [4, [0, 2, 4, 6, 8, 10, 12, 14]], [5, [0, 2, 4, 6, 8, 10, 12, 14]], [6, [0, 2, 4, 6, 8, 10, 12, 14]], [7, [0, 2, 4, 6, 8, 10, 12, 14]], [8, [0, 2, 4, 6, 8, 10, 12, 14]], [9, [0, 2, 4, 6, 8, 10, 12, 14]], [10, [0, 2, 4, 6, 8, 10, 12, 14]], [11, [0, 2, 4, 6, 8, 10, 12, 14]], [12, [0, 2, 4, 6, 8, 10, 12, 14]], [13, [0, 2, 4, 6, 8, 10, 12, 14]], [14, [0, 2, 4, 6, 8, 10, 12, 14]], [15, [0, 2, 4, 6, 8, 10, 12, 14]]], 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, 8, 10, 12, 14], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]]

After

derived = [[[0, 2, 4, 6, 8, 10, 12, 14], [1, 3, 5, 7, 9, 11, 13, 15]], [0, 1, 0, 1, 0, 1, 0, 1, 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, 8, 10, 12, 14], [0, 2, 4, 6, 8, 10, 12, 14]]

After

derived = [[[0, 0, 0, 0], [0, 2, 2, 0], [0, 4, 4, 0], [0, 6, 6, 0], [0, 8, 8, 0], [0, 10, 10, 0], [0, 12, 12, 0], [0, 14, 14, 0], [2, 0, 2, 0], [2, 2, 4, 0], [2, 4, 6, 0], [2, 6, 8, 0], [2, 8, 10, 0], [2, 10, 12, 0], [2, 12, 14, 0], [2, 14, 0, 0], [4, 0, 4, 0], [4, 2, 6, 0], [4, 4, 8, 0], [4, 6, 10, 0], [4, 8, 12, 0], [4, 10, 14, 0], [4, 12, 0, 0], [4, 14, 2, 0], [6, 0, 6, 0], [6, 2, 8, 0], [6, 4, 10, 0], [6, 6, 12, 0], [6, 8, 14, 0], [6, 10, 0, 0], [6, 12, 2, 0], [6, 14, 4, 0], [8, 0, 8, 0], [8, 2, 10, 0], [8, 4, 12, 0], [8, 6, 14, 0], [8, 8, 0, 0], [8, 10, 2, 0], [8, 12, 4, 0], [8, 14, 6, 0], [10, 0, 10, 0], [10, 2, 12, 0], [10, 4, 14, 0], [10, 6, 0, 0], [10, 8, 2, 0], [10, 10, 4, 0], [10, 12, 6, 0], [10, 14, 8, 0], [12, 0, 12, 0], [12, 2, 14, 0], [12, 4, 0, 0], [12, 6, 2, 0], [12, 8, 4, 0], [12, 10, 6, 0], [12, 12, 8, 0], [12, 14, 10, 0], [14, 0, 14, 0], [14, 2, 0, 0], [14, 4, 2, 0], [14, 6, 4, 0], [14, 8, 6, 0], [14, 10, 8, 0], [14, 12, 10, 0], [14, 14, 12, 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, 8, 10, 12, 14], [1, 3, 5, 7, 9, 11, 13, 15]]

After

derived = [[[0, 1, 1, 1], [0, 3, 3, 1], [0, 5, 5, 1], [0, 7, 7, 1], [0, 9, 9, 1], [0, 11, 11, 1], [0, 13, 13, 1], [0, 15, 15, 1], [2, 1, 3, 1], [2, 3, 5, 1], [2, 5, 7, 1], [2, 7, 9, 1], [2, 9, 11, 1], [2, 11, 13, 1], [2, 13, 15, 1], [2, 15, 1, 1], [4, 1, 5, 1], [4, 3, 7, 1], [4, 5, 9, 1], [4, 7, 11, 1], [4, 9, 13, 1], [4, 11, 15, 1], [4, 13, 1, 1], [4, 15, 3, 1], [6, 1, 7, 1], [6, 3, 9, 1], [6, 5, 11, 1], [6, 7, 13, 1], [6, 9, 15, 1], [6, 11, 1, 1], [6, 13, 3, 1], [6, 15, 5, 1], [8, 1, 9, 1], [8, 3, 11, 1], [8, 5, 13, 1], [8, 7, 15, 1], [8, 9, 1, 1], [8, 11, 3, 1], [8, 13, 5, 1], [8, 15, 7, 1], [10, 1, 11, 1], [10, 3, 13, 1], [10, 5, 15, 1], [10, 7, 1, 1], [10, 9, 3, 1], [10, 11, 5, 1], [10, 13, 7, 1], [10, 15, 9, 1], [12, 1, 13, 1], [12, 3, 15, 1], [12, 5, 1, 1], [12, 7, 3, 1], [12, 9, 5, 1], [12, 11, 7, 1], [12, 13, 9, 1], [12, 15, 11, 1], [14, 1, 15, 1], [14, 3, 1, 1], [14, 5, 3, 1], [14, 7, 5, 1], [14, 9, 7, 1], [14, 11, 9, 1], [14, 13, 11, 1], [14, 15, 13, 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.

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, 9, 11, 13, 15], [0, 2, 4, 6, 8, 10, 12, 14]]

After

derived = [[[1, 0, 1, 1], [1, 2, 3, 1], [1, 4, 5, 1], [1, 6, 7, 1], [1, 8, 9, 1], [1, 10, 11, 1], [1, 12, 13, 1], [1, 14, 15, 1], [3, 0, 3, 1], [3, 2, 5, 1], [3, 4, 7, 1], [3, 6, 9, 1], [3, 8, 11, 1], [3, 10, 13, 1], [3, 12, 15, 1], [3, 14, 1, 1], [5, 0, 5, 1], [5, 2, 7, 1], [5, 4, 9, 1], [5, 6, 11, 1], [5, 8, 13, 1], [5, 10, 15, 1], [5, 12, 1, 1], [5, 14, 3, 1], [7, 0, 7, 1], [7, 2, 9, 1], [7, 4, 11, 1], [7, 6, 13, 1], [7, 8, 15, 1], [7, 10, 1, 1], [7, 12, 3, 1], [7, 14, 5, 1], [9, 0, 9, 1], [9, 2, 11, 1], [9, 4, 13, 1], [9, 6, 15, 1], [9, 8, 1, 1], [9, 10, 3, 1], [9, 12, 5, 1], [9, 14, 7, 1], [11, 0, 11, 1], [11, 2, 13, 1], [11, 4, 15, 1], [11, 6, 1, 1], [11, 8, 3, 1], [11, 10, 5, 1], [11, 12, 7, 1], [11, 14, 9, 1], [13, 0, 13, 1], [13, 2, 15, 1], [13, 4, 1, 1], [13, 6, 3, 1], [13, 8, 5, 1], [13, 10, 7, 1], [13, 12, 9, 1], [13, 14, 11, 1], [15, 0, 15, 1], [15, 2, 1, 1], [15, 4, 3, 1], [15, 6, 5, 1], [15, 8, 7, 1], [15, 10, 9, 1], [15, 12, 11, 1], [15, 14, 13, 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.

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, 9, 11, 13, 15], [1, 3, 5, 7, 9, 11, 13, 15]]

After

derived = [[[1, 1, 2, 0], [1, 3, 4, 0], [1, 5, 6, 0], [1, 7, 8, 0], [1, 9, 10, 0], [1, 11, 12, 0], [1, 13, 14, 0], [1, 15, 0, 0], [3, 1, 4, 0], [3, 3, 6, 0], [3, 5, 8, 0], [3, 7, 10, 0], [3, 9, 12, 0], [3, 11, 14, 0], [3, 13, 0, 0], [3, 15, 2, 0], [5, 1, 6, 0], [5, 3, 8, 0], [5, 5, 10, 0], [5, 7, 12, 0], [5, 9, 14, 0], [5, 11, 0, 0], [5, 13, 2, 0], [5, 15, 4, 0], [7, 1, 8, 0], [7, 3, 10, 0], [7, 5, 12, 0], [7, 7, 14, 0], [7, 9, 0, 0], [7, 11, 2, 0], [7, 13, 4, 0], [7, 15, 6, 0], [9, 1, 10, 0], [9, 3, 12, 0], [9, 5, 14, 0], [9, 7, 0, 0], [9, 9, 2, 0], [9, 11, 4, 0], [9, 13, 6, 0], [9, 15, 8, 0], [11, 1, 12, 0], [11, 3, 14, 0], [11, 5, 0, 0], [11, 7, 2, 0], [11, 9, 4, 0], [11, 11, 6, 0], [11, 13, 8, 0], [11, 15, 10, 0], [13, 1, 14, 0], [13, 3, 0, 0], [13, 5, 2, 0], [13, 7, 4, 0], [13, 9, 6, 0], [13, 11, 8, 0], [13, 13, 10, 0], [13, 15, 12, 0], [15, 1, 0, 0], [15, 3, 2, 0], [15, 5, 4, 0], [15, 7, 6, 0], [15, 9, 8, 0], [15, 11, 10, 0], [15, 13, 12, 0], [15, 15, 14, 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, S008

S010 · implies

Before

source roles = [[0, 2, 4, 6, 8, 10, 12, 14], Yes, [[0, 2, 4, 6, 8, 10, 12, 14], [1, 3, 5, 7, 9, 11, 13, 15]]]

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, 8, 10, 12, 14], Yes, [[0, 2, 4, 6, 8, 10, 12, 14], [1, 3, 5, 7, 9, 11, 13, 15]], Yes, 2, quotient_elements_are_disjoint_complete_coset_sets_not_representatives)

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: c530e31bea38130e36620e12b845042223af56a5e1dcca052a934a40a41bc382
  • candidate record SHA-256: bd554efcce27c1596a22ab953f0702fa6bb1a5c5971f00e530943675a8895f92
  • semantic review receipt SHA-256: 6212b8b0b3b273f3a514e26ebd17d7e81219c56a8f47e40047ee2b6bf01d1ba3
  • question-first witness SHA-256: f32a23d5e1b66c7794c60def58741bcdfacff24f3cdfc9bc48109c8f71543cf3
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON