# Question 25394: Every Representative Pair Must Land in the Same Declared Product Coset [Return to catalogue page 127](https://ocme.evemisslab.com/en/questions/page/127/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261114-r02-finite-quotient-group-foundations/#q-20261015-advance-20261114-r02-finite-quotient-group-0069) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0049-25394 · q-20261015-advance-20261114-r02-finite-quotient-group-0069 ## 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, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]], identity=0, explicit subset=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38] 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, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]], identity=0, explicit subset=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38] 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](https://ocme.evemisslab.com/en/mko/mko-finite-quotient-group/) ## Step-by-step derivation S001 · impliesBefore 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 dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · impliesBefore 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, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28], [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], [31, 32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], [32, 33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], [33, 34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], [34, 35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33], [35, 36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], [36, 37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], [37, 38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36], [38, 39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [39, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]], 0] After derived = [40, [[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], [24, 24, 24, 24], [25, 25, 25, 25], [26, 26, 26, 26], [27, 27, 27, 27], [28, 28, 28, 28], [29, 29, 29, 29], [30, 30, 30, 30], [31, 31, 31, 31], [32, 32, 32, 32], [33, 33, 33, 33], [34, 34, 34, 34], [35, 35, 35, 35], [36, 36, 36, 36], [37, 37, 37, 37], [38, 38, 38, 38], [39, 39, 39, 39]], Yes] Conditions: none Validate the complete closed Cayley table and actual identity label,without assuming0 or the old32-element cap. Step identifiers and dependenciesRule: Quotient_complete_group_table_actual_identity · Uses: G001, S001 S003 · impliesBefore source roles = [40, 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, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], 64000, [0, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 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 dependenciesRule: Quotient_ALL_group_associativity_triples_and_two_sided_inverses · Uses: G001, S002 S004 · impliesBefore source roles = [0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]] 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], [0, 16, 16, Yes], [0, 18, 18, Yes], [0, 20, 20, Yes], [0, 22, 22, Yes], [0, 24, 24, Yes], [0, 26, 26, Yes], [0, 28, 28, Yes], [0, 30, 30, Yes], [0, 32, 32, Yes], [0, 34, 34, Yes], [0, 36, 36, Yes], [0, 38, 38, 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, 24, Yes], [2, 24, 26, Yes], [2, 26, 28, Yes], [2, 28, 30, Yes], [2, 30, 32, Yes], [2, 32, 34, Yes], [2, 34, 36, Yes], [2, 36, 38, Yes], [2, 38, 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, 24, Yes], [4, 22, 26, Yes], [4, 24, 28, Yes], [4, 26, 30, Yes], [4, 28, 32, Yes], [4, 30, 34, Yes], [4, 32, 36, Yes], [4, 34, 38, Yes], [4, 36, 0, Yes], [4, 38, 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, 24, Yes], [6, 20, 26, Yes], [6, 22, 28, Yes], [6, 24, 30, Yes], [6, 26, 32, Yes], [6, 28, 34, Yes], [6, 30, 36, Yes], [6, 32, 38, Yes], [6, 34, 0, Yes], [6, 36, 2, Yes], [6, 38, 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, 24, Yes], [8, 18, 26, Yes], [8, 20, 28, Yes], [8, 22, 30, Yes], [8, 24, 32, Yes], [8, 26, 34, Yes], [8, 28, 36, Yes], [8, 30, 38, Yes], [8, 32, 0, Yes], [8, 34, 2, Yes], [8, 36, 4, Yes], [8, 38, 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, 24, Yes], [10, 16, 26, Yes], [10, 18, 28, Yes], [10, 20, 30, Yes], [10, 22, 32, Yes], [10, 24, 34, Yes], [10, 26, 36, Yes], [10, 28, 38, Yes], [10, 30, 0, Yes], [10, 32, 2, Yes], [10, 34, 4, Yes], [10, 36, 6, Yes], [10, 38, 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, 24, Yes], [12, 14, 26, Yes], [12, 16, 28, Yes], [12, 18, 30, Yes], [12, 20, 32, Yes], [12, 22, 34, Yes], [12, 24, 36, Yes], [12, 26, 38, Yes], [12, 28, 0, Yes], [12, 30, 2, Yes], [12, 32, 4, Yes], [12, 34, 6, Yes], [12, 36, 8, Yes], [12, 38, 10, Yes], [14, 0, 14, Yes], [14, 2, 16, Yes], [14, 4, 18, Yes], [14, 6, 20, Yes], [14, 8, 22, Yes], [14, 10, 24, Yes], [14, 12, 26, Yes], [14, 14, 28, Yes], [14, 16, 30, Yes], [14, 18, 32, Yes], [14, 20, 34, Yes], [14, 22, 36, Yes], [14, 24, 38, Yes], [14, 26, 0, Yes], [14, 28, 2, Yes], [14, 30, 4, Yes], [14, 32, 6, Yes], [14, 34, 8, Yes], [14, 36, 10, Yes], [14, 38, 12, Yes], [16, 0, 16, Yes], [16, 2, 18, Yes], [16, 4, 20, Yes], [16, 6, 22, Yes], [16, 8, 24, Yes], [16, 10, 26, Yes], [16, 12, 28, Yes], [16, 14, 30, Yes], [16, 16, 32, Yes], [16, 18, 34, Yes], [16, 20, 36, Yes], [16, 22, 38, Yes], [16, 24, 0, Yes], [16, 26, 2, Yes], [16, 28, 4, Yes], [16, 30, 6, Yes], [16, 32, 8, Yes], [16, 34, 10, Yes], [16, 36, 12, Yes], [16, 38, 14, Yes], [18, 0, 18, Yes], [18, 2, 20, Yes], [18, 4, 22, Yes], [18, 6, 24, Yes], [18, 8, 26, Yes], [18, 10, 28, Yes], [18, 12, 30, Yes], [18, 14, 32, Yes], [18, 16, 34, Yes], [18, 18, 36, Yes], [18, 20, 38, Yes], [18, 22, 0, Yes], [18, 24, 2, Yes], [18, 26, 4, Yes], [18, 28, 6, Yes], [18, 30, 8, Yes], [18, 32, 10, Yes], [18, 34, 12, Yes], [18, 36, 14, Yes], [18, 38, 16, Yes], [20, 0, 20, Yes], [20, 2, 22, Yes], [20, 4, 24, Yes], [20, 6, 26, Yes], [20, 8, 28, Yes], [20, 10, 30, Yes], [20, 12, 32, Yes], [20, 14, 34, Yes], [20, 16, 36, Yes], [20, 18, 38, Yes], [20, 20, 0, Yes], [20, 22, 2, Yes], [20, 24, 4, Yes], [20, 26, 6, Yes], [20, 28, 8, Yes], [20, 30, 10, Yes], [20, 32, 12, Yes], [20, 34, 14, Yes], [20, 36, 16, Yes], [20, 38, 18, Yes], [22, 0, 22, Yes], [22, 2, 24, Yes], [22, 4, 26, Yes], [22, 6, 28, Yes], [22, 8, 30, Yes], [22, 10, 32, Yes], [22, 12, 34, Yes], [22, 14, 36, Yes], [22, 16, 38, Yes], [22, 18, 0, Yes], [22, 20, 2, Yes], [22, 22, 4, Yes], [22, 24, 6, Yes], [22, 26, 8, Yes], [22, 28, 10, Yes], [22, 30, 12, Yes], [22, 32, 14, Yes], [22, 34, 16, Yes], [22, 36, 18, Yes], [22, 38, 20, Yes], [24, 0, 24, Yes], [24, 2, 26, Yes], [24, 4, 28, Yes], [24, 6, 30, Yes], [24, 8, 32, Yes], [24, 10, 34, Yes], [24, 12, 36, Yes], [24, 14, 38, Yes], [24, 16, 0, Yes], [24, 18, 2, Yes], [24, 20, 4, Yes], [24, 22, 6, Yes], [24, 24, 8, Yes], [24, 26, 10, Yes], [24, 28, 12, Yes], [24, 30, 14, Yes], [24, 32, 16, Yes], [24, 34, 18, Yes], [24, 36, 20, Yes], [24, 38, 22, Yes], [26, 0, 26, Yes], [26, 2, 28, Yes], [26, 4, 30, Yes], [26, 6, 32, Yes], [26, 8, 34, Yes], [26, 10, 36, Yes], [26, 12, 38, Yes], [26, 14, 0, Yes], [26, 16, 2, Yes], [26, 18, 4, Yes], [26, 20, 6, Yes], [26, 22, 8, Yes], [26, 24, 10, Yes], [26, 26, 12, Yes], [26, 28, 14, Yes], [26, 30, 16, Yes], [26, 32, 18, Yes], [26, 34, 20, Yes], [26, 36, 22, Yes], [26, 38, 24, Yes], [28, 0, 28, Yes], [28, 2, 30, Yes], [28, 4, 32, Yes], [28, 6, 34, Yes], [28, 8, 36, Yes], [28, 10, 38, Yes], [28, 12, 0, Yes], [28, 14, 2, Yes], [28, 16, 4, Yes], [28, 18, 6, Yes], [28, 20, 8, Yes], [28, 22, 10, Yes], [28, 24, 12, Yes], [28, 26, 14, Yes], [28, 28, 16, Yes], [28, 30, 18, Yes], [28, 32, 20, Yes], [28, 34, 22, Yes], [28, 36, 24, Yes], [28, 38, 26, Yes], [30, 0, 30, Yes], [30, 2, 32, Yes], [30, 4, 34, Yes], [30, 6, 36, Yes], [30, 8, 38, Yes], [30, 10, 0, Yes], [30, 12, 2, Yes], [30, 14, 4, Yes], [30, 16, 6, Yes], [30, 18, 8, Yes], [30, 20, 10, Yes], [30, 22, 12, Yes], [30, 24, 14, Yes], [30, 26, 16, Yes], [30, 28, 18, Yes], [30, 30, 20, Yes], [30, 32, 22, Yes], [30, 34, 24, Yes], [30, 36, 26, Yes], [30, 38, 28, Yes], [32, 0, 32, Yes], [32, 2, 34, Yes], [32, 4, 36, Yes], [32, 6, 38, Yes], [32, 8, 0, Yes], [32, 10, 2, Yes], [32, 12, 4, Yes], [32, 14, 6, Yes], [32, 16, 8, Yes], [32, 18, 10, Yes], [32, 20, 12, Yes], [32, 22, 14, Yes], [32, 24, 16, Yes], [32, 26, 18, Yes], [32, 28, 20, Yes], [32, 30, 22, Yes], [32, 32, 24, Yes], [32, 34, 26, Yes], [32, 36, 28, Yes], [32, 38, 30, Yes], [34, 0, 34, Yes], [34, 2, 36, Yes], [34, 4, 38, Yes], [34, 6, 0, Yes], [34, 8, 2, Yes], [34, 10, 4, Yes], [34, 12, 6, Yes], [34, 14, 8, Yes], [34, 16, 10, Yes], [34, 18, 12, Yes], [34, 20, 14, Yes], [34, 22, 16, Yes], [34, 24, 18, Yes], [34, 26, 20, Yes], [34, 28, 22, Yes], [34, 30, 24, Yes], [34, 32, 26, Yes], [34, 34, 28, Yes], [34, 36, 30, Yes], [34, 38, 32, Yes], [36, 0, 36, Yes], [36, 2, 38, Yes], [36, 4, 0, Yes], [36, 6, 2, Yes], [36, 8, 4, Yes], [36, 10, 6, Yes], [36, 12, 8, Yes], [36, 14, 10, Yes], [36, 16, 12, Yes], [36, 18, 14, Yes], [36, 20, 16, Yes], [36, 22, 18, Yes], [36, 24, 20, Yes], [36, 26, 22, Yes], [36, 28, 24, Yes], [36, 30, 26, Yes], [36, 32, 28, Yes], [36, 34, 30, Yes], [36, 36, 32, Yes], [36, 38, 34, Yes], [38, 0, 38, Yes], [38, 2, 0, Yes], [38, 4, 2, Yes], [38, 6, 4, Yes], [38, 8, 6, Yes], [38, 10, 8, Yes], [38, 12, 10, Yes], [38, 14, 12, Yes], [38, 16, 14, Yes], [38, 18, 16, Yes], [38, 20, 18, Yes], [38, 22, 20, Yes], [38, 24, 22, Yes], [38, 26, 24, Yes], [38, 28, 26, Yes], [38, 30, 28, Yes], [38, 32, 30, Yes], [38, 34, 32, Yes], [38, 36, 34, Yes], [38, 38, 36, Yes]], [[0, 0, Yes], [2, 38, Yes], [4, 36, Yes], [6, 34, Yes], [8, 32, Yes], [10, 30, Yes], [12, 28, Yes], [14, 26, Yes], [16, 24, Yes], [18, 22, Yes], [20, 20, Yes], [22, 18, Yes], [24, 16, Yes], [26, 14, Yes], [28, 12, Yes], [30, 10, Yes], [32, 8, Yes], [34, 6, Yes], [36, 4, Yes], [38, 2, Yes]], Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]]], 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 dependenciesRule: Quotient_genuine_subgroup_before_actual_normality · Uses: G001, S003 S005 · impliesBefore source roles = [[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39]] After derived = [[[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39]], [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 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] 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 dependenciesRule: Quotient_ALL_LEFT_coset_SETS_complete_disjoint_partition · Uses: G001, S004 S006 · impliesBefore source roles = [0, 0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]] 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], [0, 16, 16, 0], [0, 18, 18, 0], [0, 20, 20, 0], [0, 22, 22, 0], [0, 24, 24, 0], [0, 26, 26, 0], [0, 28, 28, 0], [0, 30, 30, 0], [0, 32, 32, 0], [0, 34, 34, 0], [0, 36, 36, 0], [0, 38, 38, 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, 24, 0], [2, 24, 26, 0], [2, 26, 28, 0], [2, 28, 30, 0], [2, 30, 32, 0], [2, 32, 34, 0], [2, 34, 36, 0], [2, 36, 38, 0], [2, 38, 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, 24, 0], [4, 22, 26, 0], [4, 24, 28, 0], [4, 26, 30, 0], [4, 28, 32, 0], [4, 30, 34, 0], [4, 32, 36, 0], [4, 34, 38, 0], [4, 36, 0, 0], [4, 38, 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, 24, 0], [6, 20, 26, 0], [6, 22, 28, 0], [6, 24, 30, 0], [6, 26, 32, 0], [6, 28, 34, 0], [6, 30, 36, 0], [6, 32, 38, 0], [6, 34, 0, 0], [6, 36, 2, 0], [6, 38, 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, 24, 0], [8, 18, 26, 0], [8, 20, 28, 0], [8, 22, 30, 0], [8, 24, 32, 0], [8, 26, 34, 0], [8, 28, 36, 0], [8, 30, 38, 0], [8, 32, 0, 0], [8, 34, 2, 0], [8, 36, 4, 0], [8, 38, 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, 24, 0], [10, 16, 26, 0], [10, 18, 28, 0], [10, 20, 30, 0], [10, 22, 32, 0], [10, 24, 34, 0], [10, 26, 36, 0], [10, 28, 38, 0], [10, 30, 0, 0], [10, 32, 2, 0], [10, 34, 4, 0], [10, 36, 6, 0], [10, 38, 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, 24, 0], [12, 14, 26, 0], [12, 16, 28, 0], [12, 18, 30, 0], [12, 20, 32, 0], [12, 22, 34, 0], [12, 24, 36, 0], [12, 26, 38, 0], [12, 28, 0, 0], [12, 30, 2, 0], [12, 32, 4, 0], [12, 34, 6, 0], [12, 36, 8, 0], [12, 38, 10, 0], [14, 0, 14, 0], [14, 2, 16, 0], [14, 4, 18, 0], [14, 6, 20, 0], [14, 8, 22, 0], [14, 10, 24, 0], [14, 12, 26, 0], [14, 14, 28, 0], [14, 16, 30, 0], [14, 18, 32, 0], [14, 20, 34, 0], [14, 22, 36, 0], [14, 24, 38, 0], [14, 26, 0, 0], [14, 28, 2, 0], [14, 30, 4, 0], [14, 32, 6, 0], [14, 34, 8, 0], [14, 36, 10, 0], [14, 38, 12, 0], [16, 0, 16, 0], [16, 2, 18, 0], [16, 4, 20, 0], [16, 6, 22, 0], [16, 8, 24, 0], [16, 10, 26, 0], [16, 12, 28, 0], [16, 14, 30, 0], [16, 16, 32, 0], [16, 18, 34, 0], [16, 20, 36, 0], [16, 22, 38, 0], [16, 24, 0, 0], [16, 26, 2, 0], [16, 28, 4, 0], [16, 30, 6, 0], [16, 32, 8, 0], [16, 34, 10, 0], [16, 36, 12, 0], [16, 38, 14, 0], [18, 0, 18, 0], [18, 2, 20, 0], [18, 4, 22, 0], [18, 6, 24, 0], [18, 8, 26, 0], [18, 10, 28, 0], [18, 12, 30, 0], [18, 14, 32, 0], [18, 16, 34, 0], [18, 18, 36, 0], [18, 20, 38, 0], [18, 22, 0, 0], [18, 24, 2, 0], [18, 26, 4, 0], [18, 28, 6, 0], [18, 30, 8, 0], [18, 32, 10, 0], [18, 34, 12, 0], [18, 36, 14, 0], [18, 38, 16, 0], [20, 0, 20, 0], [20, 2, 22, 0], [20, 4, 24, 0], [20, 6, 26, 0], [20, 8, 28, 0], [20, 10, 30, 0], [20, 12, 32, 0], [20, 14, 34, 0], [20, 16, 36, 0], [20, 18, 38, 0], [20, 20, 0, 0], [20, 22, 2, 0], [20, 24, 4, 0], [20, 26, 6, 0], [20, 28, 8, 0], [20, 30, 10, 0], [20, 32, 12, 0], [20, 34, 14, 0], [20, 36, 16, 0], [20, 38, 18, 0], [22, 0, 22, 0], [22, 2, 24, 0], [22, 4, 26, 0], [22, 6, 28, 0], [22, 8, 30, 0], [22, 10, 32, 0], [22, 12, 34, 0], [22, 14, 36, 0], [22, 16, 38, 0], [22, 18, 0, 0], [22, 20, 2, 0], [22, 22, 4, 0], [22, 24, 6, 0], [22, 26, 8, 0], [22, 28, 10, 0], [22, 30, 12, 0], [22, 32, 14, 0], [22, 34, 16, 0], [22, 36, 18, 0], [22, 38, 20, 0], [24, 0, 24, 0], [24, 2, 26, 0], [24, 4, 28, 0], [24, 6, 30, 0], [24, 8, 32, 0], [24, 10, 34, 0], [24, 12, 36, 0], [24, 14, 38, 0], [24, 16, 0, 0], [24, 18, 2, 0], [24, 20, 4, 0], [24, 22, 6, 0], [24, 24, 8, 0], [24, 26, 10, 0], [24, 28, 12, 0], [24, 30, 14, 0], [24, 32, 16, 0], [24, 34, 18, 0], [24, 36, 20, 0], [24, 38, 22, 0], [26, 0, 26, 0], [26, 2, 28, 0], [26, 4, 30, 0], [26, 6, 32, 0], [26, 8, 34, 0], [26, 10, 36, 0], [26, 12, 38, 0], [26, 14, 0, 0], [26, 16, 2, 0], [26, 18, 4, 0], [26, 20, 6, 0], [26, 22, 8, 0], [26, 24, 10, 0], [26, 26, 12, 0], [26, 28, 14, 0], [26, 30, 16, 0], [26, 32, 18, 0], [26, 34, 20, 0], [26, 36, 22, 0], [26, 38, 24, 0], [28, 0, 28, 0], [28, 2, 30, 0], [28, 4, 32, 0], [28, 6, 34, 0], [28, 8, 36, 0], [28, 10, 38, 0], [28, 12, 0, 0], [28, 14, 2, 0], [28, 16, 4, 0], [28, 18, 6, 0], [28, 20, 8, 0], [28, 22, 10, 0], [28, 24, 12, 0], [28, 26, 14, 0], [28, 28, 16, 0], [28, 30, 18, 0], [28, 32, 20, 0], [28, 34, 22, 0], [28, 36, 24, 0], [28, 38, 26, 0], [30, 0, 30, 0], [30, 2, 32, 0], [30, 4, 34, 0], [30, 6, 36, 0], [30, 8, 38, 0], [30, 10, 0, 0], [30, 12, 2, 0], [30, 14, 4, 0], [30, 16, 6, 0], [30, 18, 8, 0], [30, 20, 10, 0], [30, 22, 12, 0], [30, 24, 14, 0], [30, 26, 16, 0], [30, 28, 18, 0], [30, 30, 20, 0], [30, 32, 22, 0], [30, 34, 24, 0], [30, 36, 26, 0], [30, 38, 28, 0], [32, 0, 32, 0], [32, 2, 34, 0], [32, 4, 36, 0], [32, 6, 38, 0], [32, 8, 0, 0], [32, 10, 2, 0], [32, 12, 4, 0], [32, 14, 6, 0], [32, 16, 8, 0], [32, 18, 10, 0], [32, 20, 12, 0], [32, 22, 14, 0], [32, 24, 16, 0], [32, 26, 18, 0], [32, 28, 20, 0], [32, 30, 22, 0], [32, 32, 24, 0], [32, 34, 26, 0], [32, 36, 28, 0], [32, 38, 30, 0], [34, 0, 34, 0], [34, 2, 36, 0], [34, 4, 38, 0], [34, 6, 0, 0], [34, 8, 2, 0], [34, 10, 4, 0], [34, 12, 6, 0], [34, 14, 8, 0], [34, 16, 10, 0], [34, 18, 12, 0], [34, 20, 14, 0], [34, 22, 16, 0], [34, 24, 18, 0], [34, 26, 20, 0], [34, 28, 22, 0], [34, 30, 24, 0], [34, 32, 26, 0], [34, 34, 28, 0], [34, 36, 30, 0], [34, 38, 32, 0], [36, 0, 36, 0], [36, 2, 38, 0], [36, 4, 0, 0], [36, 6, 2, 0], [36, 8, 4, 0], [36, 10, 6, 0], [36, 12, 8, 0], [36, 14, 10, 0], [36, 16, 12, 0], [36, 18, 14, 0], [36, 20, 16, 0], [36, 22, 18, 0], [36, 24, 20, 0], [36, 26, 22, 0], [36, 28, 24, 0], [36, 30, 26, 0], [36, 32, 28, 0], [36, 34, 30, 0], [36, 36, 32, 0], [36, 38, 34, 0], [38, 0, 38, 0], [38, 2, 0, 0], [38, 4, 2, 0], [38, 6, 4, 0], [38, 8, 6, 0], [38, 10, 8, 0], [38, 12, 10, 0], [38, 14, 12, 0], [38, 16, 14, 0], [38, 18, 16, 0], [38, 20, 18, 0], [38, 22, 20, 0], [38, 24, 22, 0], [38, 26, 24, 0], [38, 28, 26, 0], [38, 30, 28, 0], [38, 32, 30, 0], [38, 34, 32, 0], [38, 36, 34, 0], [38, 38, 36, 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 dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S005 S007 · impliesBefore source roles = [0, 1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39]] 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], [0, 17, 17, 1], [0, 19, 19, 1], [0, 21, 21, 1], [0, 23, 23, 1], [0, 25, 25, 1], [0, 27, 27, 1], [0, 29, 29, 1], [0, 31, 31, 1], [0, 33, 33, 1], [0, 35, 35, 1], [0, 37, 37, 1], [0, 39, 39, 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, 25, 1], [2, 25, 27, 1], [2, 27, 29, 1], [2, 29, 31, 1], [2, 31, 33, 1], [2, 33, 35, 1], [2, 35, 37, 1], [2, 37, 39, 1], [2, 39, 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, 25, 1], [4, 23, 27, 1], [4, 25, 29, 1], [4, 27, 31, 1], [4, 29, 33, 1], [4, 31, 35, 1], [4, 33, 37, 1], [4, 35, 39, 1], [4, 37, 1, 1], [4, 39, 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, 25, 1], [6, 21, 27, 1], [6, 23, 29, 1], [6, 25, 31, 1], [6, 27, 33, 1], [6, 29, 35, 1], [6, 31, 37, 1], [6, 33, 39, 1], [6, 35, 1, 1], [6, 37, 3, 1], [6, 39, 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, 25, 1], [8, 19, 27, 1], [8, 21, 29, 1], [8, 23, 31, 1], [8, 25, 33, 1], [8, 27, 35, 1], [8, 29, 37, 1], [8, 31, 39, 1], [8, 33, 1, 1], [8, 35, 3, 1], [8, 37, 5, 1], [8, 39, 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, 25, 1], [10, 17, 27, 1], [10, 19, 29, 1], [10, 21, 31, 1], [10, 23, 33, 1], [10, 25, 35, 1], [10, 27, 37, 1], [10, 29, 39, 1], [10, 31, 1, 1], [10, 33, 3, 1], [10, 35, 5, 1], [10, 37, 7, 1], [10, 39, 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, 25, 1], [12, 15, 27, 1], [12, 17, 29, 1], [12, 19, 31, 1], [12, 21, 33, 1], [12, 23, 35, 1], [12, 25, 37, 1], [12, 27, 39, 1], [12, 29, 1, 1], [12, 31, 3, 1], [12, 33, 5, 1], [12, 35, 7, 1], [12, 37, 9, 1], [12, 39, 11, 1], [14, 1, 15, 1], [14, 3, 17, 1], [14, 5, 19, 1], [14, 7, 21, 1], [14, 9, 23, 1], [14, 11, 25, 1], [14, 13, 27, 1], [14, 15, 29, 1], [14, 17, 31, 1], [14, 19, 33, 1], [14, 21, 35, 1], [14, 23, 37, 1], [14, 25, 39, 1], [14, 27, 1, 1], [14, 29, 3, 1], [14, 31, 5, 1], [14, 33, 7, 1], [14, 35, 9, 1], [14, 37, 11, 1], [14, 39, 13, 1], [16, 1, 17, 1], [16, 3, 19, 1], [16, 5, 21, 1], [16, 7, 23, 1], [16, 9, 25, 1], [16, 11, 27, 1], [16, 13, 29, 1], [16, 15, 31, 1], [16, 17, 33, 1], [16, 19, 35, 1], [16, 21, 37, 1], [16, 23, 39, 1], [16, 25, 1, 1], [16, 27, 3, 1], [16, 29, 5, 1], [16, 31, 7, 1], [16, 33, 9, 1], [16, 35, 11, 1], [16, 37, 13, 1], [16, 39, 15, 1], [18, 1, 19, 1], [18, 3, 21, 1], [18, 5, 23, 1], [18, 7, 25, 1], [18, 9, 27, 1], [18, 11, 29, 1], [18, 13, 31, 1], [18, 15, 33, 1], [18, 17, 35, 1], [18, 19, 37, 1], [18, 21, 39, 1], [18, 23, 1, 1], [18, 25, 3, 1], [18, 27, 5, 1], [18, 29, 7, 1], [18, 31, 9, 1], [18, 33, 11, 1], [18, 35, 13, 1], [18, 37, 15, 1], [18, 39, 17, 1], [20, 1, 21, 1], [20, 3, 23, 1], [20, 5, 25, 1], [20, 7, 27, 1], [20, 9, 29, 1], [20, 11, 31, 1], [20, 13, 33, 1], [20, 15, 35, 1], [20, 17, 37, 1], [20, 19, 39, 1], [20, 21, 1, 1], [20, 23, 3, 1], [20, 25, 5, 1], [20, 27, 7, 1], [20, 29, 9, 1], [20, 31, 11, 1], [20, 33, 13, 1], [20, 35, 15, 1], [20, 37, 17, 1], [20, 39, 19, 1], [22, 1, 23, 1], [22, 3, 25, 1], [22, 5, 27, 1], [22, 7, 29, 1], [22, 9, 31, 1], [22, 11, 33, 1], [22, 13, 35, 1], [22, 15, 37, 1], [22, 17, 39, 1], [22, 19, 1, 1], [22, 21, 3, 1], [22, 23, 5, 1], [22, 25, 7, 1], [22, 27, 9, 1], [22, 29, 11, 1], [22, 31, 13, 1], [22, 33, 15, 1], [22, 35, 17, 1], [22, 37, 19, 1], [22, 39, 21, 1], [24, 1, 25, 1], [24, 3, 27, 1], [24, 5, 29, 1], [24, 7, 31, 1], [24, 9, 33, 1], [24, 11, 35, 1], [24, 13, 37, 1], [24, 15, 39, 1], [24, 17, 1, 1], [24, 19, 3, 1], [24, 21, 5, 1], [24, 23, 7, 1], [24, 25, 9, 1], [24, 27, 11, 1], [24, 29, 13, 1], [24, 31, 15, 1], [24, 33, 17, 1], [24, 35, 19, 1], [24, 37, 21, 1], [24, 39, 23, 1], [26, 1, 27, 1], [26, 3, 29, 1], [26, 5, 31, 1], [26, 7, 33, 1], [26, 9, 35, 1], [26, 11, 37, 1], [26, 13, 39, 1], [26, 15, 1, 1], [26, 17, 3, 1], [26, 19, 5, 1], [26, 21, 7, 1], [26, 23, 9, 1], [26, 25, 11, 1], [26, 27, 13, 1], [26, 29, 15, 1], [26, 31, 17, 1], [26, 33, 19, 1], [26, 35, 21, 1], [26, 37, 23, 1], [26, 39, 25, 1], [28, 1, 29, 1], [28, 3, 31, 1], [28, 5, 33, 1], [28, 7, 35, 1], [28, 9, 37, 1], [28, 11, 39, 1], [28, 13, 1, 1], [28, 15, 3, 1], [28, 17, 5, 1], [28, 19, 7, 1], [28, 21, 9, 1], [28, 23, 11, 1], [28, 25, 13, 1], [28, 27, 15, 1], [28, 29, 17, 1], [28, 31, 19, 1], [28, 33, 21, 1], [28, 35, 23, 1], [28, 37, 25, 1], [28, 39, 27, 1], [30, 1, 31, 1], [30, 3, 33, 1], [30, 5, 35, 1], [30, 7, 37, 1], [30, 9, 39, 1], [30, 11, 1, 1], [30, 13, 3, 1], [30, 15, 5, 1], [30, 17, 7, 1], [30, 19, 9, 1], [30, 21, 11, 1], [30, 23, 13, 1], [30, 25, 15, 1], [30, 27, 17, 1], [30, 29, 19, 1], [30, 31, 21, 1], [30, 33, 23, 1], [30, 35, 25, 1], [30, 37, 27, 1], [30, 39, 29, 1], [32, 1, 33, 1], [32, 3, 35, 1], [32, 5, 37, 1], [32, 7, 39, 1], [32, 9, 1, 1], [32, 11, 3, 1], [32, 13, 5, 1], [32, 15, 7, 1], [32, 17, 9, 1], [32, 19, 11, 1], [32, 21, 13, 1], [32, 23, 15, 1], [32, 25, 17, 1], [32, 27, 19, 1], [32, 29, 21, 1], [32, 31, 23, 1], [32, 33, 25, 1], [32, 35, 27, 1], [32, 37, 29, 1], [32, 39, 31, 1], [34, 1, 35, 1], [34, 3, 37, 1], [34, 5, 39, 1], [34, 7, 1, 1], [34, 9, 3, 1], [34, 11, 5, 1], [34, 13, 7, 1], [34, 15, 9, 1], [34, 17, 11, 1], [34, 19, 13, 1], [34, 21, 15, 1], [34, 23, 17, 1], [34, 25, 19, 1], [34, 27, 21, 1], [34, 29, 23, 1], [34, 31, 25, 1], [34, 33, 27, 1], [34, 35, 29, 1], [34, 37, 31, 1], [34, 39, 33, 1], [36, 1, 37, 1], [36, 3, 39, 1], [36, 5, 1, 1], [36, 7, 3, 1], [36, 9, 5, 1], [36, 11, 7, 1], [36, 13, 9, 1], [36, 15, 11, 1], [36, 17, 13, 1], [36, 19, 15, 1], [36, 21, 17, 1], [36, 23, 19, 1], [36, 25, 21, 1], [36, 27, 23, 1], [36, 29, 25, 1], [36, 31, 27, 1], [36, 33, 29, 1], [36, 35, 31, 1], [36, 37, 33, 1], [36, 39, 35, 1], [38, 1, 39, 1], [38, 3, 1, 1], [38, 5, 3, 1], [38, 7, 5, 1], [38, 9, 7, 1], [38, 11, 9, 1], [38, 13, 11, 1], [38, 15, 13, 1], [38, 17, 15, 1], [38, 19, 17, 1], [38, 21, 19, 1], [38, 23, 21, 1], [38, 25, 23, 1], [38, 27, 25, 1], [38, 29, 27, 1], [38, 31, 29, 1], [38, 33, 31, 1], [38, 35, 33, 1], [38, 37, 35, 1], [38, 39, 37, 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 dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S006 S008 · impliesBefore source roles = [1, 0, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38]] 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], [1, 16, 17, 1], [1, 18, 19, 1], [1, 20, 21, 1], [1, 22, 23, 1], [1, 24, 25, 1], [1, 26, 27, 1], [1, 28, 29, 1], [1, 30, 31, 1], [1, 32, 33, 1], [1, 34, 35, 1], [1, 36, 37, 1], [1, 38, 39, 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, 25, 1], [3, 24, 27, 1], [3, 26, 29, 1], [3, 28, 31, 1], [3, 30, 33, 1], [3, 32, 35, 1], [3, 34, 37, 1], [3, 36, 39, 1], [3, 38, 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, 25, 1], [5, 22, 27, 1], [5, 24, 29, 1], [5, 26, 31, 1], [5, 28, 33, 1], [5, 30, 35, 1], [5, 32, 37, 1], [5, 34, 39, 1], [5, 36, 1, 1], [5, 38, 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, 25, 1], [7, 20, 27, 1], [7, 22, 29, 1], [7, 24, 31, 1], [7, 26, 33, 1], [7, 28, 35, 1], [7, 30, 37, 1], [7, 32, 39, 1], [7, 34, 1, 1], [7, 36, 3, 1], [7, 38, 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, 25, 1], [9, 18, 27, 1], [9, 20, 29, 1], [9, 22, 31, 1], [9, 24, 33, 1], [9, 26, 35, 1], [9, 28, 37, 1], [9, 30, 39, 1], [9, 32, 1, 1], [9, 34, 3, 1], [9, 36, 5, 1], [9, 38, 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, 25, 1], [11, 16, 27, 1], [11, 18, 29, 1], [11, 20, 31, 1], [11, 22, 33, 1], [11, 24, 35, 1], [11, 26, 37, 1], [11, 28, 39, 1], [11, 30, 1, 1], [11, 32, 3, 1], [11, 34, 5, 1], [11, 36, 7, 1], [11, 38, 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, 25, 1], [13, 14, 27, 1], [13, 16, 29, 1], [13, 18, 31, 1], [13, 20, 33, 1], [13, 22, 35, 1], [13, 24, 37, 1], [13, 26, 39, 1], [13, 28, 1, 1], [13, 30, 3, 1], [13, 32, 5, 1], [13, 34, 7, 1], [13, 36, 9, 1], [13, 38, 11, 1], [15, 0, 15, 1], [15, 2, 17, 1], [15, 4, 19, 1], [15, 6, 21, 1], [15, 8, 23, 1], [15, 10, 25, 1], [15, 12, 27, 1], [15, 14, 29, 1], [15, 16, 31, 1], [15, 18, 33, 1], [15, 20, 35, 1], [15, 22, 37, 1], [15, 24, 39, 1], [15, 26, 1, 1], [15, 28, 3, 1], [15, 30, 5, 1], [15, 32, 7, 1], [15, 34, 9, 1], [15, 36, 11, 1], [15, 38, 13, 1], [17, 0, 17, 1], [17, 2, 19, 1], [17, 4, 21, 1], [17, 6, 23, 1], [17, 8, 25, 1], [17, 10, 27, 1], [17, 12, 29, 1], [17, 14, 31, 1], [17, 16, 33, 1], [17, 18, 35, 1], [17, 20, 37, 1], [17, 22, 39, 1], [17, 24, 1, 1], [17, 26, 3, 1], [17, 28, 5, 1], [17, 30, 7, 1], [17, 32, 9, 1], [17, 34, 11, 1], [17, 36, 13, 1], [17, 38, 15, 1], [19, 0, 19, 1], [19, 2, 21, 1], [19, 4, 23, 1], [19, 6, 25, 1], [19, 8, 27, 1], [19, 10, 29, 1], [19, 12, 31, 1], [19, 14, 33, 1], [19, 16, 35, 1], [19, 18, 37, 1], [19, 20, 39, 1], [19, 22, 1, 1], [19, 24, 3, 1], [19, 26, 5, 1], [19, 28, 7, 1], [19, 30, 9, 1], [19, 32, 11, 1], [19, 34, 13, 1], [19, 36, 15, 1], [19, 38, 17, 1], [21, 0, 21, 1], [21, 2, 23, 1], [21, 4, 25, 1], [21, 6, 27, 1], [21, 8, 29, 1], [21, 10, 31, 1], [21, 12, 33, 1], [21, 14, 35, 1], [21, 16, 37, 1], [21, 18, 39, 1], [21, 20, 1, 1], [21, 22, 3, 1], [21, 24, 5, 1], [21, 26, 7, 1], [21, 28, 9, 1], [21, 30, 11, 1], [21, 32, 13, 1], [21, 34, 15, 1], [21, 36, 17, 1], [21, 38, 19, 1], [23, 0, 23, 1], [23, 2, 25, 1], [23, 4, 27, 1], [23, 6, 29, 1], [23, 8, 31, 1], [23, 10, 33, 1], [23, 12, 35, 1], [23, 14, 37, 1], [23, 16, 39, 1], [23, 18, 1, 1], [23, 20, 3, 1], [23, 22, 5, 1], [23, 24, 7, 1], [23, 26, 9, 1], [23, 28, 11, 1], [23, 30, 13, 1], [23, 32, 15, 1], [23, 34, 17, 1], [23, 36, 19, 1], [23, 38, 21, 1], [25, 0, 25, 1], [25, 2, 27, 1], [25, 4, 29, 1], [25, 6, 31, 1], [25, 8, 33, 1], [25, 10, 35, 1], [25, 12, 37, 1], [25, 14, 39, 1], [25, 16, 1, 1], [25, 18, 3, 1], [25, 20, 5, 1], [25, 22, 7, 1], [25, 24, 9, 1], [25, 26, 11, 1], [25, 28, 13, 1], [25, 30, 15, 1], [25, 32, 17, 1], [25, 34, 19, 1], [25, 36, 21, 1], [25, 38, 23, 1], [27, 0, 27, 1], [27, 2, 29, 1], [27, 4, 31, 1], [27, 6, 33, 1], [27, 8, 35, 1], [27, 10, 37, 1], [27, 12, 39, 1], [27, 14, 1, 1], [27, 16, 3, 1], [27, 18, 5, 1], [27, 20, 7, 1], [27, 22, 9, 1], [27, 24, 11, 1], [27, 26, 13, 1], [27, 28, 15, 1], [27, 30, 17, 1], [27, 32, 19, 1], [27, 34, 21, 1], [27, 36, 23, 1], [27, 38, 25, 1], [29, 0, 29, 1], [29, 2, 31, 1], [29, 4, 33, 1], [29, 6, 35, 1], [29, 8, 37, 1], [29, 10, 39, 1], [29, 12, 1, 1], [29, 14, 3, 1], [29, 16, 5, 1], [29, 18, 7, 1], [29, 20, 9, 1], [29, 22, 11, 1], [29, 24, 13, 1], [29, 26, 15, 1], [29, 28, 17, 1], [29, 30, 19, 1], [29, 32, 21, 1], [29, 34, 23, 1], [29, 36, 25, 1], [29, 38, 27, 1], [31, 0, 31, 1], [31, 2, 33, 1], [31, 4, 35, 1], [31, 6, 37, 1], [31, 8, 39, 1], [31, 10, 1, 1], [31, 12, 3, 1], [31, 14, 5, 1], [31, 16, 7, 1], [31, 18, 9, 1], [31, 20, 11, 1], [31, 22, 13, 1], [31, 24, 15, 1], [31, 26, 17, 1], [31, 28, 19, 1], [31, 30, 21, 1], [31, 32, 23, 1], [31, 34, 25, 1], [31, 36, 27, 1], [31, 38, 29, 1], [33, 0, 33, 1], [33, 2, 35, 1], [33, 4, 37, 1], [33, 6, 39, 1], [33, 8, 1, 1], [33, 10, 3, 1], [33, 12, 5, 1], [33, 14, 7, 1], [33, 16, 9, 1], [33, 18, 11, 1], [33, 20, 13, 1], [33, 22, 15, 1], [33, 24, 17, 1], [33, 26, 19, 1], [33, 28, 21, 1], [33, 30, 23, 1], [33, 32, 25, 1], [33, 34, 27, 1], [33, 36, 29, 1], [33, 38, 31, 1], [35, 0, 35, 1], [35, 2, 37, 1], [35, 4, 39, 1], [35, 6, 1, 1], [35, 8, 3, 1], [35, 10, 5, 1], [35, 12, 7, 1], [35, 14, 9, 1], [35, 16, 11, 1], [35, 18, 13, 1], [35, 20, 15, 1], [35, 22, 17, 1], [35, 24, 19, 1], [35, 26, 21, 1], [35, 28, 23, 1], [35, 30, 25, 1], [35, 32, 27, 1], [35, 34, 29, 1], [35, 36, 31, 1], [35, 38, 33, 1], [37, 0, 37, 1], [37, 2, 39, 1], [37, 4, 1, 1], [37, 6, 3, 1], [37, 8, 5, 1], [37, 10, 7, 1], [37, 12, 9, 1], [37, 14, 11, 1], [37, 16, 13, 1], [37, 18, 15, 1], [37, 20, 17, 1], [37, 22, 19, 1], [37, 24, 21, 1], [37, 26, 23, 1], [37, 28, 25, 1], [37, 30, 27, 1], [37, 32, 29, 1], [37, 34, 31, 1], [37, 36, 33, 1], [37, 38, 35, 1], [39, 0, 39, 1], [39, 2, 1, 1], [39, 4, 3, 1], [39, 6, 5, 1], [39, 8, 7, 1], [39, 10, 9, 1], [39, 12, 11, 1], [39, 14, 13, 1], [39, 16, 15, 1], [39, 18, 17, 1], [39, 20, 19, 1], [39, 22, 21, 1], [39, 24, 23, 1], [39, 26, 25, 1], [39, 28, 27, 1], [39, 30, 29, 1], [39, 32, 31, 1], [39, 34, 33, 1], [39, 36, 35, 1], [39, 38, 37, 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 dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S007 S009 · impliesBefore source roles = [1, 1, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39]] 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, 16, 0], [1, 17, 18, 0], [1, 19, 20, 0], [1, 21, 22, 0], [1, 23, 24, 0], [1, 25, 26, 0], [1, 27, 28, 0], [1, 29, 30, 0], [1, 31, 32, 0], [1, 33, 34, 0], [1, 35, 36, 0], [1, 37, 38, 0], [1, 39, 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, 24, 0], [3, 23, 26, 0], [3, 25, 28, 0], [3, 27, 30, 0], [3, 29, 32, 0], [3, 31, 34, 0], [3, 33, 36, 0], [3, 35, 38, 0], [3, 37, 0, 0], [3, 39, 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, 24, 0], [5, 21, 26, 0], [5, 23, 28, 0], [5, 25, 30, 0], [5, 27, 32, 0], [5, 29, 34, 0], [5, 31, 36, 0], [5, 33, 38, 0], [5, 35, 0, 0], [5, 37, 2, 0], [5, 39, 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, 24, 0], [7, 19, 26, 0], [7, 21, 28, 0], [7, 23, 30, 0], [7, 25, 32, 0], [7, 27, 34, 0], [7, 29, 36, 0], [7, 31, 38, 0], [7, 33, 0, 0], [7, 35, 2, 0], [7, 37, 4, 0], [7, 39, 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, 24, 0], [9, 17, 26, 0], [9, 19, 28, 0], [9, 21, 30, 0], [9, 23, 32, 0], [9, 25, 34, 0], [9, 27, 36, 0], [9, 29, 38, 0], [9, 31, 0, 0], [9, 33, 2, 0], [9, 35, 4, 0], [9, 37, 6, 0], [9, 39, 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, 24, 0], [11, 15, 26, 0], [11, 17, 28, 0], [11, 19, 30, 0], [11, 21, 32, 0], [11, 23, 34, 0], [11, 25, 36, 0], [11, 27, 38, 0], [11, 29, 0, 0], [11, 31, 2, 0], [11, 33, 4, 0], [11, 35, 6, 0], [11, 37, 8, 0], [11, 39, 10, 0], [13, 1, 14, 0], [13, 3, 16, 0], [13, 5, 18, 0], [13, 7, 20, 0], [13, 9, 22, 0], [13, 11, 24, 0], [13, 13, 26, 0], [13, 15, 28, 0], [13, 17, 30, 0], [13, 19, 32, 0], [13, 21, 34, 0], [13, 23, 36, 0], [13, 25, 38, 0], [13, 27, 0, 0], [13, 29, 2, 0], [13, 31, 4, 0], [13, 33, 6, 0], [13, 35, 8, 0], [13, 37, 10, 0], [13, 39, 12, 0], [15, 1, 16, 0], [15, 3, 18, 0], [15, 5, 20, 0], [15, 7, 22, 0], [15, 9, 24, 0], [15, 11, 26, 0], [15, 13, 28, 0], [15, 15, 30, 0], [15, 17, 32, 0], [15, 19, 34, 0], [15, 21, 36, 0], [15, 23, 38, 0], [15, 25, 0, 0], [15, 27, 2, 0], [15, 29, 4, 0], [15, 31, 6, 0], [15, 33, 8, 0], [15, 35, 10, 0], [15, 37, 12, 0], [15, 39, 14, 0], [17, 1, 18, 0], [17, 3, 20, 0], [17, 5, 22, 0], [17, 7, 24, 0], [17, 9, 26, 0], [17, 11, 28, 0], [17, 13, 30, 0], [17, 15, 32, 0], [17, 17, 34, 0], [17, 19, 36, 0], [17, 21, 38, 0], [17, 23, 0, 0], [17, 25, 2, 0], [17, 27, 4, 0], [17, 29, 6, 0], [17, 31, 8, 0], [17, 33, 10, 0], [17, 35, 12, 0], [17, 37, 14, 0], [17, 39, 16, 0], [19, 1, 20, 0], [19, 3, 22, 0], [19, 5, 24, 0], [19, 7, 26, 0], [19, 9, 28, 0], [19, 11, 30, 0], [19, 13, 32, 0], [19, 15, 34, 0], [19, 17, 36, 0], [19, 19, 38, 0], [19, 21, 0, 0], [19, 23, 2, 0], [19, 25, 4, 0], [19, 27, 6, 0], [19, 29, 8, 0], [19, 31, 10, 0], [19, 33, 12, 0], [19, 35, 14, 0], [19, 37, 16, 0], [19, 39, 18, 0], [21, 1, 22, 0], [21, 3, 24, 0], [21, 5, 26, 0], [21, 7, 28, 0], [21, 9, 30, 0], [21, 11, 32, 0], [21, 13, 34, 0], [21, 15, 36, 0], [21, 17, 38, 0], [21, 19, 0, 0], [21, 21, 2, 0], [21, 23, 4, 0], [21, 25, 6, 0], [21, 27, 8, 0], [21, 29, 10, 0], [21, 31, 12, 0], [21, 33, 14, 0], [21, 35, 16, 0], [21, 37, 18, 0], [21, 39, 20, 0], [23, 1, 24, 0], [23, 3, 26, 0], [23, 5, 28, 0], [23, 7, 30, 0], [23, 9, 32, 0], [23, 11, 34, 0], [23, 13, 36, 0], [23, 15, 38, 0], [23, 17, 0, 0], [23, 19, 2, 0], [23, 21, 4, 0], [23, 23, 6, 0], [23, 25, 8, 0], [23, 27, 10, 0], [23, 29, 12, 0], [23, 31, 14, 0], [23, 33, 16, 0], [23, 35, 18, 0], [23, 37, 20, 0], [23, 39, 22, 0], [25, 1, 26, 0], [25, 3, 28, 0], [25, 5, 30, 0], [25, 7, 32, 0], [25, 9, 34, 0], [25, 11, 36, 0], [25, 13, 38, 0], [25, 15, 0, 0], [25, 17, 2, 0], [25, 19, 4, 0], [25, 21, 6, 0], [25, 23, 8, 0], [25, 25, 10, 0], [25, 27, 12, 0], [25, 29, 14, 0], [25, 31, 16, 0], [25, 33, 18, 0], [25, 35, 20, 0], [25, 37, 22, 0], [25, 39, 24, 0], [27, 1, 28, 0], [27, 3, 30, 0], [27, 5, 32, 0], [27, 7, 34, 0], [27, 9, 36, 0], [27, 11, 38, 0], [27, 13, 0, 0], [27, 15, 2, 0], [27, 17, 4, 0], [27, 19, 6, 0], [27, 21, 8, 0], [27, 23, 10, 0], [27, 25, 12, 0], [27, 27, 14, 0], [27, 29, 16, 0], [27, 31, 18, 0], [27, 33, 20, 0], [27, 35, 22, 0], [27, 37, 24, 0], [27, 39, 26, 0], [29, 1, 30, 0], [29, 3, 32, 0], [29, 5, 34, 0], [29, 7, 36, 0], [29, 9, 38, 0], [29, 11, 0, 0], [29, 13, 2, 0], [29, 15, 4, 0], [29, 17, 6, 0], [29, 19, 8, 0], [29, 21, 10, 0], [29, 23, 12, 0], [29, 25, 14, 0], [29, 27, 16, 0], [29, 29, 18, 0], [29, 31, 20, 0], [29, 33, 22, 0], [29, 35, 24, 0], [29, 37, 26, 0], [29, 39, 28, 0], [31, 1, 32, 0], [31, 3, 34, 0], [31, 5, 36, 0], [31, 7, 38, 0], [31, 9, 0, 0], [31, 11, 2, 0], [31, 13, 4, 0], [31, 15, 6, 0], [31, 17, 8, 0], [31, 19, 10, 0], [31, 21, 12, 0], [31, 23, 14, 0], [31, 25, 16, 0], [31, 27, 18, 0], [31, 29, 20, 0], [31, 31, 22, 0], [31, 33, 24, 0], [31, 35, 26, 0], [31, 37, 28, 0], [31, 39, 30, 0], [33, 1, 34, 0], [33, 3, 36, 0], [33, 5, 38, 0], [33, 7, 0, 0], [33, 9, 2, 0], [33, 11, 4, 0], [33, 13, 6, 0], [33, 15, 8, 0], [33, 17, 10, 0], [33, 19, 12, 0], [33, 21, 14, 0], [33, 23, 16, 0], [33, 25, 18, 0], [33, 27, 20, 0], [33, 29, 22, 0], [33, 31, 24, 0], [33, 33, 26, 0], [33, 35, 28, 0], [33, 37, 30, 0], [33, 39, 32, 0], [35, 1, 36, 0], [35, 3, 38, 0], [35, 5, 0, 0], [35, 7, 2, 0], [35, 9, 4, 0], [35, 11, 6, 0], [35, 13, 8, 0], [35, 15, 10, 0], [35, 17, 12, 0], [35, 19, 14, 0], [35, 21, 16, 0], [35, 23, 18, 0], [35, 25, 20, 0], [35, 27, 22, 0], [35, 29, 24, 0], [35, 31, 26, 0], [35, 33, 28, 0], [35, 35, 30, 0], [35, 37, 32, 0], [35, 39, 34, 0], [37, 1, 38, 0], [37, 3, 0, 0], [37, 5, 2, 0], [37, 7, 4, 0], [37, 9, 6, 0], [37, 11, 8, 0], [37, 13, 10, 0], [37, 15, 12, 0], [37, 17, 14, 0], [37, 19, 16, 0], [37, 21, 18, 0], [37, 23, 20, 0], [37, 25, 22, 0], [37, 27, 24, 0], [37, 29, 26, 0], [37, 31, 28, 0], [37, 33, 30, 0], [37, 35, 32, 0], [37, 37, 34, 0], [37, 39, 36, 0], [39, 1, 0, 0], [39, 3, 2, 0], [39, 5, 4, 0], [39, 7, 6, 0], [39, 9, 8, 0], [39, 11, 10, 0], [39, 13, 12, 0], [39, 15, 14, 0], [39, 17, 16, 0], [39, 19, 18, 0], [39, 21, 20, 0], [39, 23, 22, 0], [39, 25, 24, 0], [39, 27, 26, 0], [39, 29, 28, 0], [39, 31, 30, 0], [39, 33, 32, 0], [39, 35, 34, 0], [39, 37, 36, 0], [39, 39, 38, 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 dependenciesRule: Quotient_EVERY_coset_pair_ALL_representative_products · Uses: G001, S008 S010 · impliesBefore source roles = [[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], Yes, [[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39]]] 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 dependenciesRule: 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, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38], [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39]], [[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: d7d062c38b4dc62b262667534a4fee2c84f0f2def74b88158d67376ed107abfb - candidate record SHA-256: 671a4e63ad120a953704a9ad74b9158c22a89cb15a0ca1a4519a2b9d9fd4c66d - semantic review receipt SHA-256: 6212b8b0b3b273f3a514e26ebd17d7e81219c56a8f47e40047ee2b6bf01d1ba3 - question-first witness SHA-256: f32a23d5e1b66c7794c60def58741bcdfacff24f3cdfc9bc48109c8f71543cf3 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261114-r02-finite-quotient-group-0069.json)