Question 25311: Quotient Products Must Be Representative Independent

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, 16, 17, 18, 19, 20, 21, 22, 23], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]], identity=0, explicit subset=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22] 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, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]], identity=0, explicit subset=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22] 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).

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

source roles = [quotient-coset-multiplication-table]

derived = [[cosets, multiplication_table, identity_coset, inverse_rows, well_defined, commutativity_certificate, certificate]]

source roles = [[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]], 0]

derived = [24, [[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], [16, 16, 16, 16], [17, 17, 17, 17], [18, 18, 18, 18], [19, 19, 19, 19], [20, 20, 20, 20], [21, 21, 21, 21], [22, 22, 22, 22], [23, 23, 23, 23]], Yes]

source roles = [24, 0]

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, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], 13824, [0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]]

source roles = [0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22]]

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], [0, 16, 16, Yes], [0, 18, 18, Yes], [0, 20, 20, Yes], [0, 22, 22, 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, 16, Yes], [2, 16, 18, Yes], [2, 18, 20, Yes], [2, 20, 22, Yes], [2, 22, 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, 16, Yes], [4, 14, 18, Yes], [4, 16, 20, Yes], [4, 18, 22, Yes], [4, 20, 0, Yes], [4, 22, 2, Yes], [6, 0, 6, Yes], [6, 2, 8, Yes], [6, 4, 10, Yes], [6, 6, 12, Yes], [6, 8, 14, Yes], [6, 10, 16, Yes], [6, 12, 18, Yes], [6, 14, 20, Yes], [6, 16, 22, Yes], [6, 18, 0, Yes], [6, 20, 2, Yes], [6, 22, 4, Yes], [8, 0, 8, Yes], [8, 2, 10, Yes], [8, 4, 12, Yes], [8, 6, 14, Yes], [8, 8, 16, Yes], [8, 10, 18, Yes], [8, 12, 20, Yes], [8, 14, 22, Yes], [8, 16, 0, Yes], [8, 18, 2, Yes], [8, 20, 4, Yes], [8, 22, 6, Yes], [10, 0, 10, Yes], [10, 2, 12, Yes], [10, 4, 14, Yes], [10, 6, 16, Yes], [10, 8, 18, Yes], [10, 10, 20, Yes], [10, 12, 22, Yes], [10, 14, 0, Yes], [10, 16, 2, Yes], [10, 18, 4, Yes], [10, 20, 6, Yes], [10, 22, 8, Yes], [12, 0, 12, Yes], [12, 2, 14, Yes], [12, 4, 16, Yes], [12, 6, 18, Yes], [12, 8, 20, Yes], [12, 10, 22, Yes], [12, 12, 0, Yes], [12, 14, 2, Yes], [12, 16, 4, Yes], [12, 18, 6, Yes], [12, 20, 8, Yes], [12, 22, 10, Yes], [14, 0, 14, Yes], [14, 2, 16, Yes], [14, 4, 18, Yes], [14, 6, 20, Yes], [14, 8, 22, Yes], [14, 10, 0, Yes], [14, 12, 2, Yes], [14, 14, 4, Yes], [14, 16, 6, Yes], [14, 18, 8, Yes], [14, 20, 10, Yes], [14, 22, 12, Yes], [16, 0, 16, Yes], [16, 2, 18, Yes], [16, 4, 20, Yes], [16, 6, 22, Yes], [16, 8, 0, Yes], [16, 10, 2, Yes], [16, 12, 4, Yes], [16, 14, 6, Yes], [16, 16, 8, Yes], [16, 18, 10, Yes], [16, 20, 12, Yes], [16, 22, 14, Yes], [18, 0, 18, Yes], [18, 2, 20, Yes], [18, 4, 22, Yes], [18, 6, 0, Yes], [18, 8, 2, Yes], [18, 10, 4, Yes], [18, 12, 6, Yes], [18, 14, 8, Yes], [18, 16, 10, Yes], [18, 18, 12, Yes], [18, 20, 14, Yes], [18, 22, 16, Yes], [20, 0, 20, Yes], [20, 2, 22, Yes], [20, 4, 0, Yes], [20, 6, 2, Yes], [20, 8, 4, Yes], [20, 10, 6, Yes], [20, 12, 8, Yes], [20, 14, 10, Yes], [20, 16, 12, Yes], [20, 18, 14, Yes], [20, 20, 16, Yes], [20, 22, 18, Yes], [22, 0, 22, Yes], [22, 2, 0, Yes], [22, 4, 2, Yes], [22, 6, 4, Yes], [22, 8, 6, Yes], [22, 10, 8, Yes], [22, 12, 10, Yes], [22, 14, 12, Yes], [22, 16, 14, Yes], [22, 18, 16, Yes], [22, 20, 18, Yes], [22, 22, 20, Yes]], [[0, 0, Yes], [2, 22, Yes], [4, 20, Yes], [6, 18, Yes], [8, 16, Yes], [10, 14, Yes], [12, 12, Yes], [14, 10, Yes], [16, 8, Yes], [18, 6, Yes], [20, 4, Yes], [22, 2, Yes]], Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22]], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22]], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22]], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22]], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22]], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22]], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22]], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22]], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22]], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22]], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22]], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22]], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22]], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22]], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22]], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22]], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22]], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22]], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22]], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22]], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22]], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22]], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22]], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22]]], Yes]

source roles = [[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]]

derived = [[[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23]], [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], Yes]

source roles = [0, 0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22]]

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], [0, 16, 16, 0], [0, 18, 18, 0], [0, 20, 20, 0], [0, 22, 22, 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, 16, 0], [2, 16, 18, 0], [2, 18, 20, 0], [2, 20, 22, 0], [2, 22, 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, 16, 0], [4, 14, 18, 0], [4, 16, 20, 0], [4, 18, 22, 0], [4, 20, 0, 0], [4, 22, 2, 0], [6, 0, 6, 0], [6, 2, 8, 0], [6, 4, 10, 0], [6, 6, 12, 0], [6, 8, 14, 0], [6, 10, 16, 0], [6, 12, 18, 0], [6, 14, 20, 0], [6, 16, 22, 0], [6, 18, 0, 0], [6, 20, 2, 0], [6, 22, 4, 0], [8, 0, 8, 0], [8, 2, 10, 0], [8, 4, 12, 0], [8, 6, 14, 0], [8, 8, 16, 0], [8, 10, 18, 0], [8, 12, 20, 0], [8, 14, 22, 0], [8, 16, 0, 0], [8, 18, 2, 0], [8, 20, 4, 0], [8, 22, 6, 0], [10, 0, 10, 0], [10, 2, 12, 0], [10, 4, 14, 0], [10, 6, 16, 0], [10, 8, 18, 0], [10, 10, 20, 0], [10, 12, 22, 0], [10, 14, 0, 0], [10, 16, 2, 0], [10, 18, 4, 0], [10, 20, 6, 0], [10, 22, 8, 0], [12, 0, 12, 0], [12, 2, 14, 0], [12, 4, 16, 0], [12, 6, 18, 0], [12, 8, 20, 0], [12, 10, 22, 0], [12, 12, 0, 0], [12, 14, 2, 0], [12, 16, 4, 0], [12, 18, 6, 0], [12, 20, 8, 0], [12, 22, 10, 0], [14, 0, 14, 0], [14, 2, 16, 0], [14, 4, 18, 0], [14, 6, 20, 0], [14, 8, 22, 0], [14, 10, 0, 0], [14, 12, 2, 0], [14, 14, 4, 0], [14, 16, 6, 0], [14, 18, 8, 0], [14, 20, 10, 0], [14, 22, 12, 0], [16, 0, 16, 0], [16, 2, 18, 0], [16, 4, 20, 0], [16, 6, 22, 0], [16, 8, 0, 0], [16, 10, 2, 0], [16, 12, 4, 0], [16, 14, 6, 0], [16, 16, 8, 0], [16, 18, 10, 0], [16, 20, 12, 0], [16, 22, 14, 0], [18, 0, 18, 0], [18, 2, 20, 0], [18, 4, 22, 0], [18, 6, 0, 0], [18, 8, 2, 0], [18, 10, 4, 0], [18, 12, 6, 0], [18, 14, 8, 0], [18, 16, 10, 0], [18, 18, 12, 0], [18, 20, 14, 0], [18, 22, 16, 0], [20, 0, 20, 0], [20, 2, 22, 0], [20, 4, 0, 0], [20, 6, 2, 0], [20, 8, 4, 0], [20, 10, 6, 0], [20, 12, 8, 0], [20, 14, 10, 0], [20, 16, 12, 0], [20, 18, 14, 0], [20, 20, 16, 0], [20, 22, 18, 0], [22, 0, 22, 0], [22, 2, 0, 0], [22, 4, 2, 0], [22, 6, 4, 0], [22, 8, 6, 0], [22, 10, 8, 0], [22, 12, 10, 0], [22, 14, 12, 0], [22, 16, 14, 0], [22, 18, 16, 0], [22, 20, 18, 0], [22, 22, 20, 0]], [0], Yes]

source roles = [0, 1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23]]

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], [0, 17, 17, 1], [0, 19, 19, 1], [0, 21, 21, 1], [0, 23, 23, 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, 17, 1], [2, 17, 19, 1], [2, 19, 21, 1], [2, 21, 23, 1], [2, 23, 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, 17, 1], [4, 15, 19, 1], [4, 17, 21, 1], [4, 19, 23, 1], [4, 21, 1, 1], [4, 23, 3, 1], [6, 1, 7, 1], [6, 3, 9, 1], [6, 5, 11, 1], [6, 7, 13, 1], [6, 9, 15, 1], [6, 11, 17, 1], [6, 13, 19, 1], [6, 15, 21, 1], [6, 17, 23, 1], [6, 19, 1, 1], [6, 21, 3, 1], [6, 23, 5, 1], [8, 1, 9, 1], [8, 3, 11, 1], [8, 5, 13, 1], [8, 7, 15, 1], [8, 9, 17, 1], [8, 11, 19, 1], [8, 13, 21, 1], [8, 15, 23, 1], [8, 17, 1, 1], [8, 19, 3, 1], [8, 21, 5, 1], [8, 23, 7, 1], [10, 1, 11, 1], [10, 3, 13, 1], [10, 5, 15, 1], [10, 7, 17, 1], [10, 9, 19, 1], [10, 11, 21, 1], [10, 13, 23, 1], [10, 15, 1, 1], [10, 17, 3, 1], [10, 19, 5, 1], [10, 21, 7, 1], [10, 23, 9, 1], [12, 1, 13, 1], [12, 3, 15, 1], [12, 5, 17, 1], [12, 7, 19, 1], [12, 9, 21, 1], [12, 11, 23, 1], [12, 13, 1, 1], [12, 15, 3, 1], [12, 17, 5, 1], [12, 19, 7, 1], [12, 21, 9, 1], [12, 23, 11, 1], [14, 1, 15, 1], [14, 3, 17, 1], [14, 5, 19, 1], [14, 7, 21, 1], [14, 9, 23, 1], [14, 11, 1, 1], [14, 13, 3, 1], [14, 15, 5, 1], [14, 17, 7, 1], [14, 19, 9, 1], [14, 21, 11, 1], [14, 23, 13, 1], [16, 1, 17, 1], [16, 3, 19, 1], [16, 5, 21, 1], [16, 7, 23, 1], [16, 9, 1, 1], [16, 11, 3, 1], [16, 13, 5, 1], [16, 15, 7, 1], [16, 17, 9, 1], [16, 19, 11, 1], [16, 21, 13, 1], [16, 23, 15, 1], [18, 1, 19, 1], [18, 3, 21, 1], [18, 5, 23, 1], [18, 7, 1, 1], [18, 9, 3, 1], [18, 11, 5, 1], [18, 13, 7, 1], [18, 15, 9, 1], [18, 17, 11, 1], [18, 19, 13, 1], [18, 21, 15, 1], [18, 23, 17, 1], [20, 1, 21, 1], [20, 3, 23, 1], [20, 5, 1, 1], [20, 7, 3, 1], [20, 9, 5, 1], [20, 11, 7, 1], [20, 13, 9, 1], [20, 15, 11, 1], [20, 17, 13, 1], [20, 19, 15, 1], [20, 21, 17, 1], [20, 23, 19, 1], [22, 1, 23, 1], [22, 3, 1, 1], [22, 5, 3, 1], [22, 7, 5, 1], [22, 9, 7, 1], [22, 11, 9, 1], [22, 13, 11, 1], [22, 15, 13, 1], [22, 17, 15, 1], [22, 19, 17, 1], [22, 21, 19, 1], [22, 23, 21, 1]], [1], Yes]

source roles = [1, 0, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22]]

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], [1, 16, 17, 1], [1, 18, 19, 1], [1, 20, 21, 1], [1, 22, 23, 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, 17, 1], [3, 16, 19, 1], [3, 18, 21, 1], [3, 20, 23, 1], [3, 22, 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, 17, 1], [5, 14, 19, 1], [5, 16, 21, 1], [5, 18, 23, 1], [5, 20, 1, 1], [5, 22, 3, 1], [7, 0, 7, 1], [7, 2, 9, 1], [7, 4, 11, 1], [7, 6, 13, 1], [7, 8, 15, 1], [7, 10, 17, 1], [7, 12, 19, 1], [7, 14, 21, 1], [7, 16, 23, 1], [7, 18, 1, 1], [7, 20, 3, 1], [7, 22, 5, 1], [9, 0, 9, 1], [9, 2, 11, 1], [9, 4, 13, 1], [9, 6, 15, 1], [9, 8, 17, 1], [9, 10, 19, 1], [9, 12, 21, 1], [9, 14, 23, 1], [9, 16, 1, 1], [9, 18, 3, 1], [9, 20, 5, 1], [9, 22, 7, 1], [11, 0, 11, 1], [11, 2, 13, 1], [11, 4, 15, 1], [11, 6, 17, 1], [11, 8, 19, 1], [11, 10, 21, 1], [11, 12, 23, 1], [11, 14, 1, 1], [11, 16, 3, 1], [11, 18, 5, 1], [11, 20, 7, 1], [11, 22, 9, 1], [13, 0, 13, 1], [13, 2, 15, 1], [13, 4, 17, 1], [13, 6, 19, 1], [13, 8, 21, 1], [13, 10, 23, 1], [13, 12, 1, 1], [13, 14, 3, 1], [13, 16, 5, 1], [13, 18, 7, 1], [13, 20, 9, 1], [13, 22, 11, 1], [15, 0, 15, 1], [15, 2, 17, 1], [15, 4, 19, 1], [15, 6, 21, 1], [15, 8, 23, 1], [15, 10, 1, 1], [15, 12, 3, 1], [15, 14, 5, 1], [15, 16, 7, 1], [15, 18, 9, 1], [15, 20, 11, 1], [15, 22, 13, 1], [17, 0, 17, 1], [17, 2, 19, 1], [17, 4, 21, 1], [17, 6, 23, 1], [17, 8, 1, 1], [17, 10, 3, 1], [17, 12, 5, 1], [17, 14, 7, 1], [17, 16, 9, 1], [17, 18, 11, 1], [17, 20, 13, 1], [17, 22, 15, 1], [19, 0, 19, 1], [19, 2, 21, 1], [19, 4, 23, 1], [19, 6, 1, 1], [19, 8, 3, 1], [19, 10, 5, 1], [19, 12, 7, 1], [19, 14, 9, 1], [19, 16, 11, 1], [19, 18, 13, 1], [19, 20, 15, 1], [19, 22, 17, 1], [21, 0, 21, 1], [21, 2, 23, 1], [21, 4, 1, 1], [21, 6, 3, 1], [21, 8, 5, 1], [21, 10, 7, 1], [21, 12, 9, 1], [21, 14, 11, 1], [21, 16, 13, 1], [21, 18, 15, 1], [21, 20, 17, 1], [21, 22, 19, 1], [23, 0, 23, 1], [23, 2, 1, 1], [23, 4, 3, 1], [23, 6, 5, 1], [23, 8, 7, 1], [23, 10, 9, 1], [23, 12, 11, 1], [23, 14, 13, 1], [23, 16, 15, 1], [23, 18, 17, 1], [23, 20, 19, 1], [23, 22, 21, 1]], [1], Yes]

source roles = [1, 1, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23]]

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, 16, 0], [1, 17, 18, 0], [1, 19, 20, 0], [1, 21, 22, 0], [1, 23, 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, 16, 0], [3, 15, 18, 0], [3, 17, 20, 0], [3, 19, 22, 0], [3, 21, 0, 0], [3, 23, 2, 0], [5, 1, 6, 0], [5, 3, 8, 0], [5, 5, 10, 0], [5, 7, 12, 0], [5, 9, 14, 0], [5, 11, 16, 0], [5, 13, 18, 0], [5, 15, 20, 0], [5, 17, 22, 0], [5, 19, 0, 0], [5, 21, 2, 0], [5, 23, 4, 0], [7, 1, 8, 0], [7, 3, 10, 0], [7, 5, 12, 0], [7, 7, 14, 0], [7, 9, 16, 0], [7, 11, 18, 0], [7, 13, 20, 0], [7, 15, 22, 0], [7, 17, 0, 0], [7, 19, 2, 0], [7, 21, 4, 0], [7, 23, 6, 0], [9, 1, 10, 0], [9, 3, 12, 0], [9, 5, 14, 0], [9, 7, 16, 0], [9, 9, 18, 0], [9, 11, 20, 0], [9, 13, 22, 0], [9, 15, 0, 0], [9, 17, 2, 0], [9, 19, 4, 0], [9, 21, 6, 0], [9, 23, 8, 0], [11, 1, 12, 0], [11, 3, 14, 0], [11, 5, 16, 0], [11, 7, 18, 0], [11, 9, 20, 0], [11, 11, 22, 0], [11, 13, 0, 0], [11, 15, 2, 0], [11, 17, 4, 0], [11, 19, 6, 0], [11, 21, 8, 0], [11, 23, 10, 0], [13, 1, 14, 0], [13, 3, 16, 0], [13, 5, 18, 0], [13, 7, 20, 0], [13, 9, 22, 0], [13, 11, 0, 0], [13, 13, 2, 0], [13, 15, 4, 0], [13, 17, 6, 0], [13, 19, 8, 0], [13, 21, 10, 0], [13, 23, 12, 0], [15, 1, 16, 0], [15, 3, 18, 0], [15, 5, 20, 0], [15, 7, 22, 0], [15, 9, 0, 0], [15, 11, 2, 0], [15, 13, 4, 0], [15, 15, 6, 0], [15, 17, 8, 0], [15, 19, 10, 0], [15, 21, 12, 0], [15, 23, 14, 0], [17, 1, 18, 0], [17, 3, 20, 0], [17, 5, 22, 0], [17, 7, 0, 0], [17, 9, 2, 0], [17, 11, 4, 0], [17, 13, 6, 0], [17, 15, 8, 0], [17, 17, 10, 0], [17, 19, 12, 0], [17, 21, 14, 0], [17, 23, 16, 0], [19, 1, 20, 0], [19, 3, 22, 0], [19, 5, 0, 0], [19, 7, 2, 0], [19, 9, 4, 0], [19, 11, 6, 0], [19, 13, 8, 0], [19, 15, 10, 0], [19, 17, 12, 0], [19, 19, 14, 0], [19, 21, 16, 0], [19, 23, 18, 0], [21, 1, 22, 0], [21, 3, 0, 0], [21, 5, 2, 0], [21, 7, 4, 0], [21, 9, 6, 0], [21, 11, 8, 0], [21, 13, 10, 0], [21, 15, 12, 0], [21, 17, 14, 0], [21, 19, 16, 0], [21, 21, 18, 0], [21, 23, 20, 0], [23, 1, 0, 0], [23, 3, 2, 0], [23, 5, 4, 0], [23, 7, 6, 0], [23, 9, 8, 0], [23, 11, 10, 0], [23, 13, 12, 0], [23, 15, 14, 0], [23, 17, 16, 0], [23, 19, 18, 0], [23, 21, 20, 0], [23, 23, 22, 0]], [0], Yes]

source roles = [[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], Yes, [[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23]]]

derived = [[[0, 1], [1, 0]], 0, [[0, 0], [1, 1]], Yes, No]

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, 16, 18, 20, 22], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23]], [[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.

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.