# Question 25748: First Isomorphism Reaches the Image, Not Automatically the Whole Codomain [Return to catalogue page 129](https://ocme.evemisslab.com/en/questions/page/129/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations/#q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0098) 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. R0050-25748 · q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0098 ## Original question and givens Use complete source Cayley 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, target 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, 40, 41, 42, 43, 44, 45, 46, 47], [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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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], [40, 41, 42, 43, 44, 45, 46, 47, 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], [41, 42, 43, 44, 45, 46, 47, 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, 40], [42, 43, 44, 45, 46, 47, 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, 40, 41], [43, 44, 45, 46, 47, 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, 40, 41, 42], [44, 45, 46, 47, 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, 40, 41, 42, 43], [45, 46, 47, 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, 40, 41, 42, 43, 44], [46, 47, 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, 40, 41, 42, 43, 44, 45], [47, 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, 40, 41, 42, 43, 44, 45, 46]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary). - G001 — Use complete source Cayley 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, target 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, 40, 41, 42, 43, 44, 45, 46, 47], [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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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], [40, 41, 42, 43, 44, 45, 46, 47, 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], [41, 42, 43, 44, 45, 46, 47, 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, 40], [42, 43, 44, 45, 46, 47, 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, 40, 41], [43, 44, 45, 46, 47, 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, 40, 41, 42], [44, 45, 46, 47, 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, 40, 41, 42, 43], [45, 46, 47, 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, 40, 41, 42, 43, 44], [46, 47, 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, 40, 41, 42, 43, 44, 45], [47, 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, 40, 41, 42, 43, 44, 45, 46]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary). ## 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-first-isomorphism-theorem-finite-groups](https://ocme.evemisslab.com/en/mko/mko-first-isomorphism-theorem-finite-groups/) ## Step-by-step derivation S001 · impliesBefore source roles = [iso1-image-codomain-boundary] After derived = [[image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary]] Conditions: none List the full source field order without JavaScript.LP weak uses13 fields,zero/strong15;premises and roles follow the original operation/mode without padding. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · impliesBefore source roles = [Iso1 ORIGINAL SOURCE, [[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] After 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] Conditions: none Qualify the complete closed Cayley table and actual identity first,allowing order64 without assuming identity0. Step identifiers and dependenciesRule: Group_complete_table_actual_identity_bound64 · Uses: G001, S001 S003 · impliesBefore source roles = [Iso1 ORIGINAL SOURCE, 24, 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]], 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]] Conditions: none Each pair-matrix entry checks every third element for associativity,covering all n³ triples,not samples.Inverses return the actual identity from both sides. Step identifiers and dependenciesRule: Group_ALL_triples_compact_pair_matrix_and_ALL_inverses · Uses: G001, S002 S004 · impliesBefore source roles = [Iso1 ORIGINAL TARGET, [[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, 40, 41, 42, 43, 44, 45, 46, 47], [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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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, 40, 41, 42, 43, 44, 45, 46, 47, 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], [40, 41, 42, 43, 44, 45, 46, 47, 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], [41, 42, 43, 44, 45, 46, 47, 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, 40], [42, 43, 44, 45, 46, 47, 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, 40, 41], [43, 44, 45, 46, 47, 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, 40, 41, 42], [44, 45, 46, 47, 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, 40, 41, 42, 43], [45, 46, 47, 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, 40, 41, 42, 43, 44], [46, 47, 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, 40, 41, 42, 43, 44, 45], [47, 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, 40, 41, 42, 43, 44, 45, 46]], 0] After derived = [48, [[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], [40, 40, 40, 40], [41, 41, 41, 41], [42, 42, 42, 42], [43, 43, 43, 43], [44, 44, 44, 44], [45, 45, 45, 45], [46, 46, 46, 46], [47, 47, 47, 47]], Yes] Conditions: none Qualify the complete closed Cayley table and actual identity first,allowing order64 without assuming identity0. Step identifiers and dependenciesRule: Group_complete_table_actual_identity_bound64 · Uses: G001, S003 S005 · impliesBefore source roles = [Iso1 ORIGINAL TARGET, 48, 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, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes]], 110592, [0, 47, 46, 45, 44, 43, 42, 41, 40, 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 every third element for associativity,covering all n³ triples,not samples.Inverses return the actual identity from both sides. Step identifiers and dependenciesRule: Group_ALL_triples_compact_pair_matrix_and_ALL_inverses · Uses: G001, S004 S006 · impliesBefore source roles = [Iso1 ACTUAL kernel, [0], 0] After derived = [Yes, [[0, 0, 0, Yes]], Yes, [[0, 0, Yes]], Yes, Yes, [[0, [0], Yes], [1, [0], Yes], [2, [0], Yes], [3, [0], Yes], [4, [0], Yes], [5, [0], Yes], [6, [0], Yes], [7, [0], Yes], [8, [0], Yes], [9, [0], Yes], [10, [0], Yes], [11, [0], Yes], [12, [0], Yes], [13, [0], Yes], [14, [0], Yes], [15, [0], Yes], [16, [0], Yes], [17, [0], Yes], [18, [0], Yes], [19, [0], Yes], [20, [0], Yes], [21, [0], Yes], [22, [0], Yes], [23, [0], Yes]], Yes] Conditions: none Check identity,all products and inverse closure separately before subgroup qualification,then every g for normality.These are actual properties that may individually hold even when a map fails;do not fill flags from a theorem name. Step identifiers and dependenciesRule: Subgroup_actual_identity_product_inverse_ALL_conjugates · Uses: G001, S005 S007 · impliesBefore source roles = [Iso1 ACTUAL image, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 0] After derived = [Yes, [[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], [0, 40, 40, Yes], [0, 42, 42, Yes], [0, 44, 44, Yes], [0, 46, 46, 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, 40, Yes], [2, 40, 42, Yes], [2, 42, 44, Yes], [2, 44, 46, Yes], [2, 46, 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, 40, Yes], [4, 38, 42, Yes], [4, 40, 44, Yes], [4, 42, 46, Yes], [4, 44, 0, Yes], [4, 46, 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, 40, Yes], [6, 36, 42, Yes], [6, 38, 44, Yes], [6, 40, 46, Yes], [6, 42, 0, Yes], [6, 44, 2, Yes], [6, 46, 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, 40, Yes], [8, 34, 42, Yes], [8, 36, 44, Yes], [8, 38, 46, Yes], [8, 40, 0, Yes], [8, 42, 2, Yes], [8, 44, 4, Yes], [8, 46, 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, 40, Yes], [10, 32, 42, Yes], [10, 34, 44, Yes], [10, 36, 46, Yes], [10, 38, 0, Yes], [10, 40, 2, Yes], [10, 42, 4, Yes], [10, 44, 6, Yes], [10, 46, 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, 40, Yes], [12, 30, 42, Yes], [12, 32, 44, Yes], [12, 34, 46, Yes], [12, 36, 0, Yes], [12, 38, 2, Yes], [12, 40, 4, Yes], [12, 42, 6, Yes], [12, 44, 8, Yes], [12, 46, 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, 40, Yes], [14, 28, 42, Yes], [14, 30, 44, Yes], [14, 32, 46, Yes], [14, 34, 0, Yes], [14, 36, 2, Yes], [14, 38, 4, Yes], [14, 40, 6, Yes], [14, 42, 8, Yes], [14, 44, 10, Yes], [14, 46, 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, 40, Yes], [16, 26, 42, Yes], [16, 28, 44, Yes], [16, 30, 46, Yes], [16, 32, 0, Yes], [16, 34, 2, Yes], [16, 36, 4, Yes], [16, 38, 6, Yes], [16, 40, 8, Yes], [16, 42, 10, Yes], [16, 44, 12, Yes], [16, 46, 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, 40, Yes], [18, 24, 42, Yes], [18, 26, 44, Yes], [18, 28, 46, Yes], [18, 30, 0, Yes], [18, 32, 2, Yes], [18, 34, 4, Yes], [18, 36, 6, Yes], [18, 38, 8, Yes], [18, 40, 10, Yes], [18, 42, 12, Yes], [18, 44, 14, Yes], [18, 46, 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, 40, Yes], [20, 22, 42, Yes], [20, 24, 44, Yes], [20, 26, 46, Yes], [20, 28, 0, Yes], [20, 30, 2, Yes], [20, 32, 4, Yes], [20, 34, 6, Yes], [20, 36, 8, Yes], [20, 38, 10, Yes], [20, 40, 12, Yes], [20, 42, 14, Yes], [20, 44, 16, Yes], [20, 46, 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, 40, Yes], [22, 20, 42, Yes], [22, 22, 44, Yes], [22, 24, 46, Yes], [22, 26, 0, Yes], [22, 28, 2, Yes], [22, 30, 4, Yes], [22, 32, 6, Yes], [22, 34, 8, Yes], [22, 36, 10, Yes], [22, 38, 12, Yes], [22, 40, 14, Yes], [22, 42, 16, Yes], [22, 44, 18, Yes], [22, 46, 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, 40, Yes], [24, 18, 42, Yes], [24, 20, 44, Yes], [24, 22, 46, Yes], [24, 24, 0, Yes], [24, 26, 2, Yes], [24, 28, 4, Yes], [24, 30, 6, Yes], [24, 32, 8, Yes], [24, 34, 10, Yes], [24, 36, 12, Yes], [24, 38, 14, Yes], [24, 40, 16, Yes], [24, 42, 18, Yes], [24, 44, 20, Yes], [24, 46, 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, 40, Yes], [26, 16, 42, Yes], [26, 18, 44, Yes], [26, 20, 46, Yes], [26, 22, 0, Yes], [26, 24, 2, Yes], [26, 26, 4, Yes], [26, 28, 6, Yes], [26, 30, 8, Yes], [26, 32, 10, Yes], [26, 34, 12, Yes], [26, 36, 14, Yes], [26, 38, 16, Yes], [26, 40, 18, Yes], [26, 42, 20, Yes], [26, 44, 22, Yes], [26, 46, 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, 40, Yes], [28, 14, 42, Yes], [28, 16, 44, Yes], [28, 18, 46, Yes], [28, 20, 0, Yes], [28, 22, 2, Yes], [28, 24, 4, Yes], [28, 26, 6, Yes], [28, 28, 8, Yes], [28, 30, 10, Yes], [28, 32, 12, Yes], [28, 34, 14, Yes], [28, 36, 16, Yes], [28, 38, 18, Yes], [28, 40, 20, Yes], [28, 42, 22, Yes], [28, 44, 24, Yes], [28, 46, 26, Yes], [30, 0, 30, Yes], [30, 2, 32, Yes], [30, 4, 34, Yes], [30, 6, 36, Yes], [30, 8, 38, Yes], [30, 10, 40, Yes], [30, 12, 42, Yes], [30, 14, 44, Yes], [30, 16, 46, Yes], [30, 18, 0, Yes], [30, 20, 2, Yes], [30, 22, 4, Yes], [30, 24, 6, Yes], [30, 26, 8, Yes], [30, 28, 10, Yes], [30, 30, 12, Yes], [30, 32, 14, Yes], [30, 34, 16, Yes], [30, 36, 18, Yes], [30, 38, 20, Yes], [30, 40, 22, Yes], [30, 42, 24, Yes], [30, 44, 26, Yes], [30, 46, 28, Yes], [32, 0, 32, Yes], [32, 2, 34, Yes], [32, 4, 36, Yes], [32, 6, 38, Yes], [32, 8, 40, Yes], [32, 10, 42, Yes], [32, 12, 44, Yes], [32, 14, 46, Yes], [32, 16, 0, Yes], [32, 18, 2, Yes], [32, 20, 4, Yes], [32, 22, 6, Yes], [32, 24, 8, Yes], [32, 26, 10, Yes], [32, 28, 12, Yes], [32, 30, 14, Yes], [32, 32, 16, Yes], [32, 34, 18, Yes], [32, 36, 20, Yes], [32, 38, 22, Yes], [32, 40, 24, Yes], [32, 42, 26, Yes], [32, 44, 28, Yes], [32, 46, 30, Yes], [34, 0, 34, Yes], [34, 2, 36, Yes], [34, 4, 38, Yes], [34, 6, 40, Yes], [34, 8, 42, Yes], [34, 10, 44, Yes], [34, 12, 46, Yes], [34, 14, 0, Yes], [34, 16, 2, Yes], [34, 18, 4, Yes], [34, 20, 6, Yes], [34, 22, 8, Yes], [34, 24, 10, Yes], [34, 26, 12, Yes], [34, 28, 14, Yes], [34, 30, 16, Yes], [34, 32, 18, Yes], [34, 34, 20, Yes], [34, 36, 22, Yes], [34, 38, 24, Yes], [34, 40, 26, Yes], [34, 42, 28, Yes], [34, 44, 30, Yes], [34, 46, 32, Yes], [36, 0, 36, Yes], [36, 2, 38, Yes], [36, 4, 40, Yes], [36, 6, 42, Yes], [36, 8, 44, Yes], [36, 10, 46, Yes], [36, 12, 0, Yes], [36, 14, 2, Yes], [36, 16, 4, Yes], [36, 18, 6, Yes], [36, 20, 8, Yes], [36, 22, 10, Yes], [36, 24, 12, Yes], [36, 26, 14, Yes], [36, 28, 16, Yes], [36, 30, 18, Yes], [36, 32, 20, Yes], [36, 34, 22, Yes], [36, 36, 24, Yes], [36, 38, 26, Yes], [36, 40, 28, Yes], [36, 42, 30, Yes], [36, 44, 32, Yes], [36, 46, 34, Yes], [38, 0, 38, Yes], [38, 2, 40, Yes], [38, 4, 42, Yes], [38, 6, 44, Yes], [38, 8, 46, Yes], [38, 10, 0, Yes], [38, 12, 2, Yes], [38, 14, 4, Yes], [38, 16, 6, Yes], [38, 18, 8, Yes], [38, 20, 10, Yes], [38, 22, 12, Yes], [38, 24, 14, Yes], [38, 26, 16, Yes], [38, 28, 18, Yes], [38, 30, 20, Yes], [38, 32, 22, Yes], [38, 34, 24, Yes], [38, 36, 26, Yes], [38, 38, 28, Yes], [38, 40, 30, Yes], [38, 42, 32, Yes], [38, 44, 34, Yes], [38, 46, 36, Yes], [40, 0, 40, Yes], [40, 2, 42, Yes], [40, 4, 44, Yes], [40, 6, 46, Yes], [40, 8, 0, Yes], [40, 10, 2, Yes], [40, 12, 4, Yes], [40, 14, 6, Yes], [40, 16, 8, Yes], [40, 18, 10, Yes], [40, 20, 12, Yes], [40, 22, 14, Yes], [40, 24, 16, Yes], [40, 26, 18, Yes], [40, 28, 20, Yes], [40, 30, 22, Yes], [40, 32, 24, Yes], [40, 34, 26, Yes], [40, 36, 28, Yes], [40, 38, 30, Yes], [40, 40, 32, Yes], [40, 42, 34, Yes], [40, 44, 36, Yes], [40, 46, 38, Yes], [42, 0, 42, Yes], [42, 2, 44, Yes], [42, 4, 46, Yes], [42, 6, 0, Yes], [42, 8, 2, Yes], [42, 10, 4, Yes], [42, 12, 6, Yes], [42, 14, 8, Yes], [42, 16, 10, Yes], [42, 18, 12, Yes], [42, 20, 14, Yes], [42, 22, 16, Yes], [42, 24, 18, Yes], [42, 26, 20, Yes], [42, 28, 22, Yes], [42, 30, 24, Yes], [42, 32, 26, Yes], [42, 34, 28, Yes], [42, 36, 30, Yes], [42, 38, 32, Yes], [42, 40, 34, Yes], [42, 42, 36, Yes], [42, 44, 38, Yes], [42, 46, 40, Yes], [44, 0, 44, Yes], [44, 2, 46, Yes], [44, 4, 0, Yes], [44, 6, 2, Yes], [44, 8, 4, Yes], [44, 10, 6, Yes], [44, 12, 8, Yes], [44, 14, 10, Yes], [44, 16, 12, Yes], [44, 18, 14, Yes], [44, 20, 16, Yes], [44, 22, 18, Yes], [44, 24, 20, Yes], [44, 26, 22, Yes], [44, 28, 24, Yes], [44, 30, 26, Yes], [44, 32, 28, Yes], [44, 34, 30, Yes], [44, 36, 32, Yes], [44, 38, 34, Yes], [44, 40, 36, Yes], [44, 42, 38, Yes], [44, 44, 40, Yes], [44, 46, 42, Yes], [46, 0, 46, Yes], [46, 2, 0, Yes], [46, 4, 2, Yes], [46, 6, 4, Yes], [46, 8, 6, Yes], [46, 10, 8, Yes], [46, 12, 10, Yes], [46, 14, 12, Yes], [46, 16, 14, Yes], [46, 18, 16, Yes], [46, 20, 18, Yes], [46, 22, 20, Yes], [46, 24, 22, Yes], [46, 26, 24, Yes], [46, 28, 26, Yes], [46, 30, 28, Yes], [46, 32, 30, Yes], [46, 34, 32, Yes], [46, 36, 34, Yes], [46, 38, 36, Yes], [46, 40, 38, Yes], [46, 42, 40, Yes], [46, 44, 42, Yes], [46, 46, 44, Yes]], Yes, [[0, 0, Yes], [2, 46, Yes], [4, 44, Yes], [6, 42, Yes], [8, 40, Yes], [10, 38, Yes], [12, 36, Yes], [14, 34, Yes], [16, 32, Yes], [18, 30, Yes], [20, 28, Yes], [22, 26, Yes], [24, 24, Yes], [26, 22, Yes], [28, 20, Yes], [30, 18, Yes], [32, 16, Yes], [34, 14, Yes], [36, 12, Yes], [38, 10, Yes], [40, 8, Yes], [42, 6, Yes], [44, 4, Yes], [46, 2, Yes]], Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [38, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [39, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [40, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [41, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [42, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [43, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [44, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [45, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [46, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes], [47, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes]], Yes] Conditions: none Check identity,all products and inverse closure separately before subgroup qualification,then every g for normality.These are actual properties that may individually hold even when a map fails;do not fill flags from a theorem name. Step identifiers and dependenciesRule: Subgroup_actual_identity_product_inverse_ALL_conjugates · Uses: G001, S006 S008 · impliesBefore source roles = [[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 0, 0] After derived = [[[0, 0, 0, 0, Yes], [0, 1, 2, 2, Yes], [0, 2, 4, 4, Yes], [0, 3, 6, 6, Yes], [0, 4, 8, 8, Yes], [0, 5, 10, 10, Yes], [0, 6, 12, 12, Yes], [0, 7, 14, 14, Yes], [0, 8, 16, 16, Yes], [0, 9, 18, 18, Yes], [0, 10, 20, 20, Yes], [0, 11, 22, 22, Yes], [0, 12, 24, 24, Yes], [0, 13, 26, 26, Yes], [0, 14, 28, 28, Yes], [0, 15, 30, 30, Yes], [0, 16, 32, 32, Yes], [0, 17, 34, 34, Yes], [0, 18, 36, 36, Yes], [0, 19, 38, 38, Yes], [0, 20, 40, 40, Yes], [0, 21, 42, 42, Yes], [0, 22, 44, 44, Yes], [0, 23, 46, 46, Yes], [1, 0, 2, 2, Yes], [1, 1, 4, 4, Yes], [1, 2, 6, 6, Yes], [1, 3, 8, 8, Yes], [1, 4, 10, 10, Yes], [1, 5, 12, 12, Yes], [1, 6, 14, 14, Yes], [1, 7, 16, 16, Yes], [1, 8, 18, 18, Yes], [1, 9, 20, 20, Yes], [1, 10, 22, 22, Yes], [1, 11, 24, 24, Yes], [1, 12, 26, 26, Yes], [1, 13, 28, 28, Yes], [1, 14, 30, 30, Yes], [1, 15, 32, 32, Yes], [1, 16, 34, 34, Yes], [1, 17, 36, 36, Yes], [1, 18, 38, 38, Yes], [1, 19, 40, 40, Yes], [1, 20, 42, 42, Yes], [1, 21, 44, 44, Yes], [1, 22, 46, 46, Yes], [1, 23, 0, 0, Yes], [2, 0, 4, 4, Yes], [2, 1, 6, 6, Yes], [2, 2, 8, 8, Yes], [2, 3, 10, 10, Yes], [2, 4, 12, 12, Yes], [2, 5, 14, 14, Yes], [2, 6, 16, 16, Yes], [2, 7, 18, 18, Yes], [2, 8, 20, 20, Yes], [2, 9, 22, 22, Yes], [2, 10, 24, 24, Yes], [2, 11, 26, 26, Yes], [2, 12, 28, 28, Yes], [2, 13, 30, 30, Yes], [2, 14, 32, 32, Yes], [2, 15, 34, 34, Yes], [2, 16, 36, 36, Yes], [2, 17, 38, 38, Yes], [2, 18, 40, 40, Yes], [2, 19, 42, 42, Yes], [2, 20, 44, 44, Yes], [2, 21, 46, 46, Yes], [2, 22, 0, 0, Yes], [2, 23, 2, 2, Yes], [3, 0, 6, 6, Yes], [3, 1, 8, 8, Yes], [3, 2, 10, 10, Yes], [3, 3, 12, 12, Yes], [3, 4, 14, 14, Yes], [3, 5, 16, 16, Yes], [3, 6, 18, 18, Yes], [3, 7, 20, 20, Yes], [3, 8, 22, 22, Yes], [3, 9, 24, 24, Yes], [3, 10, 26, 26, Yes], [3, 11, 28, 28, Yes], [3, 12, 30, 30, Yes], [3, 13, 32, 32, Yes], [3, 14, 34, 34, Yes], [3, 15, 36, 36, Yes], [3, 16, 38, 38, Yes], [3, 17, 40, 40, Yes], [3, 18, 42, 42, Yes], [3, 19, 44, 44, Yes], [3, 20, 46, 46, Yes], [3, 21, 0, 0, Yes], [3, 22, 2, 2, Yes], [3, 23, 4, 4, Yes], [4, 0, 8, 8, Yes], [4, 1, 10, 10, Yes], [4, 2, 12, 12, Yes], [4, 3, 14, 14, Yes], [4, 4, 16, 16, Yes], [4, 5, 18, 18, Yes], [4, 6, 20, 20, Yes], [4, 7, 22, 22, Yes], [4, 8, 24, 24, Yes], [4, 9, 26, 26, Yes], [4, 10, 28, 28, Yes], [4, 11, 30, 30, Yes], [4, 12, 32, 32, Yes], [4, 13, 34, 34, Yes], [4, 14, 36, 36, Yes], [4, 15, 38, 38, Yes], [4, 16, 40, 40, Yes], [4, 17, 42, 42, Yes], [4, 18, 44, 44, Yes], [4, 19, 46, 46, Yes], [4, 20, 0, 0, Yes], [4, 21, 2, 2, Yes], [4, 22, 4, 4, Yes], [4, 23, 6, 6, Yes], [5, 0, 10, 10, Yes], [5, 1, 12, 12, Yes], [5, 2, 14, 14, Yes], [5, 3, 16, 16, Yes], [5, 4, 18, 18, Yes], [5, 5, 20, 20, Yes], [5, 6, 22, 22, Yes], [5, 7, 24, 24, Yes], [5, 8, 26, 26, Yes], [5, 9, 28, 28, Yes], [5, 10, 30, 30, Yes], [5, 11, 32, 32, Yes], [5, 12, 34, 34, Yes], [5, 13, 36, 36, Yes], [5, 14, 38, 38, Yes], [5, 15, 40, 40, Yes], [5, 16, 42, 42, Yes], [5, 17, 44, 44, Yes], [5, 18, 46, 46, Yes], [5, 19, 0, 0, Yes], [5, 20, 2, 2, Yes], [5, 21, 4, 4, Yes], [5, 22, 6, 6, Yes], [5, 23, 8, 8, Yes], [6, 0, 12, 12, Yes], [6, 1, 14, 14, Yes], [6, 2, 16, 16, Yes], [6, 3, 18, 18, Yes], [6, 4, 20, 20, Yes], [6, 5, 22, 22, Yes], [6, 6, 24, 24, Yes], [6, 7, 26, 26, Yes], [6, 8, 28, 28, Yes], [6, 9, 30, 30, Yes], [6, 10, 32, 32, Yes], [6, 11, 34, 34, Yes], [6, 12, 36, 36, Yes], [6, 13, 38, 38, Yes], [6, 14, 40, 40, Yes], [6, 15, 42, 42, Yes], [6, 16, 44, 44, Yes], [6, 17, 46, 46, Yes], [6, 18, 0, 0, Yes], [6, 19, 2, 2, Yes], [6, 20, 4, 4, Yes], [6, 21, 6, 6, Yes], [6, 22, 8, 8, Yes], [6, 23, 10, 10, Yes], [7, 0, 14, 14, Yes], [7, 1, 16, 16, Yes], [7, 2, 18, 18, Yes], [7, 3, 20, 20, Yes], [7, 4, 22, 22, Yes], [7, 5, 24, 24, Yes], [7, 6, 26, 26, Yes], [7, 7, 28, 28, Yes], [7, 8, 30, 30, Yes], [7, 9, 32, 32, Yes], [7, 10, 34, 34, Yes], [7, 11, 36, 36, Yes], [7, 12, 38, 38, Yes], [7, 13, 40, 40, Yes], [7, 14, 42, 42, Yes], [7, 15, 44, 44, Yes], [7, 16, 46, 46, Yes], [7, 17, 0, 0, Yes], [7, 18, 2, 2, Yes], [7, 19, 4, 4, Yes], [7, 20, 6, 6, Yes], [7, 21, 8, 8, Yes], [7, 22, 10, 10, Yes], [7, 23, 12, 12, Yes], [8, 0, 16, 16, Yes], [8, 1, 18, 18, Yes], [8, 2, 20, 20, Yes], [8, 3, 22, 22, Yes], [8, 4, 24, 24, Yes], [8, 5, 26, 26, Yes], [8, 6, 28, 28, Yes], [8, 7, 30, 30, Yes], [8, 8, 32, 32, Yes], [8, 9, 34, 34, Yes], [8, 10, 36, 36, Yes], [8, 11, 38, 38, Yes], [8, 12, 40, 40, Yes], [8, 13, 42, 42, Yes], [8, 14, 44, 44, Yes], [8, 15, 46, 46, Yes], [8, 16, 0, 0, Yes], [8, 17, 2, 2, Yes], [8, 18, 4, 4, Yes], [8, 19, 6, 6, Yes], [8, 20, 8, 8, Yes], [8, 21, 10, 10, Yes], [8, 22, 12, 12, Yes], [8, 23, 14, 14, Yes], [9, 0, 18, 18, Yes], [9, 1, 20, 20, Yes], [9, 2, 22, 22, Yes], [9, 3, 24, 24, Yes], [9, 4, 26, 26, Yes], [9, 5, 28, 28, Yes], [9, 6, 30, 30, Yes], [9, 7, 32, 32, Yes], [9, 8, 34, 34, Yes], [9, 9, 36, 36, Yes], [9, 10, 38, 38, Yes], [9, 11, 40, 40, Yes], [9, 12, 42, 42, Yes], [9, 13, 44, 44, Yes], [9, 14, 46, 46, Yes], [9, 15, 0, 0, Yes], [9, 16, 2, 2, Yes], [9, 17, 4, 4, Yes], [9, 18, 6, 6, Yes], [9, 19, 8, 8, Yes], [9, 20, 10, 10, Yes], [9, 21, 12, 12, Yes], [9, 22, 14, 14, Yes], [9, 23, 16, 16, Yes], [10, 0, 20, 20, Yes], [10, 1, 22, 22, Yes], [10, 2, 24, 24, Yes], [10, 3, 26, 26, Yes], [10, 4, 28, 28, Yes], [10, 5, 30, 30, Yes], [10, 6, 32, 32, Yes], [10, 7, 34, 34, Yes], [10, 8, 36, 36, Yes], [10, 9, 38, 38, Yes], [10, 10, 40, 40, Yes], [10, 11, 42, 42, Yes], [10, 12, 44, 44, Yes], [10, 13, 46, 46, Yes], [10, 14, 0, 0, Yes], [10, 15, 2, 2, Yes], [10, 16, 4, 4, Yes], [10, 17, 6, 6, Yes], [10, 18, 8, 8, Yes], [10, 19, 10, 10, Yes], [10, 20, 12, 12, Yes], [10, 21, 14, 14, Yes], [10, 22, 16, 16, Yes], [10, 23, 18, 18, Yes], [11, 0, 22, 22, Yes], [11, 1, 24, 24, Yes], [11, 2, 26, 26, Yes], [11, 3, 28, 28, Yes], [11, 4, 30, 30, Yes], [11, 5, 32, 32, Yes], [11, 6, 34, 34, Yes], [11, 7, 36, 36, Yes], [11, 8, 38, 38, Yes], [11, 9, 40, 40, Yes], [11, 10, 42, 42, Yes], [11, 11, 44, 44, Yes], [11, 12, 46, 46, Yes], [11, 13, 0, 0, Yes], [11, 14, 2, 2, Yes], [11, 15, 4, 4, Yes], [11, 16, 6, 6, Yes], [11, 17, 8, 8, Yes], [11, 18, 10, 10, Yes], [11, 19, 12, 12, Yes], [11, 20, 14, 14, Yes], [11, 21, 16, 16, Yes], [11, 22, 18, 18, Yes], [11, 23, 20, 20, Yes], [12, 0, 24, 24, Yes], [12, 1, 26, 26, Yes], [12, 2, 28, 28, Yes], [12, 3, 30, 30, Yes], [12, 4, 32, 32, Yes], [12, 5, 34, 34, Yes], [12, 6, 36, 36, Yes], [12, 7, 38, 38, Yes], [12, 8, 40, 40, Yes], [12, 9, 42, 42, Yes], [12, 10, 44, 44, Yes], [12, 11, 46, 46, Yes], [12, 12, 0, 0, Yes], [12, 13, 2, 2, Yes], [12, 14, 4, 4, Yes], [12, 15, 6, 6, Yes], [12, 16, 8, 8, Yes], [12, 17, 10, 10, Yes], [12, 18, 12, 12, Yes], [12, 19, 14, 14, Yes], [12, 20, 16, 16, Yes], [12, 21, 18, 18, Yes], [12, 22, 20, 20, Yes], [12, 23, 22, 22, Yes], [13, 0, 26, 26, Yes], [13, 1, 28, 28, Yes], [13, 2, 30, 30, Yes], [13, 3, 32, 32, Yes], [13, 4, 34, 34, Yes], [13, 5, 36, 36, Yes], [13, 6, 38, 38, Yes], [13, 7, 40, 40, Yes], [13, 8, 42, 42, Yes], [13, 9, 44, 44, Yes], [13, 10, 46, 46, Yes], [13, 11, 0, 0, Yes], [13, 12, 2, 2, Yes], [13, 13, 4, 4, Yes], [13, 14, 6, 6, Yes], [13, 15, 8, 8, Yes], [13, 16, 10, 10, Yes], [13, 17, 12, 12, Yes], [13, 18, 14, 14, Yes], [13, 19, 16, 16, Yes], [13, 20, 18, 18, Yes], [13, 21, 20, 20, Yes], [13, 22, 22, 22, Yes], [13, 23, 24, 24, Yes], [14, 0, 28, 28, Yes], [14, 1, 30, 30, Yes], [14, 2, 32, 32, Yes], [14, 3, 34, 34, Yes], [14, 4, 36, 36, Yes], [14, 5, 38, 38, Yes], [14, 6, 40, 40, Yes], [14, 7, 42, 42, Yes], [14, 8, 44, 44, Yes], [14, 9, 46, 46, Yes], [14, 10, 0, 0, Yes], [14, 11, 2, 2, Yes], [14, 12, 4, 4, Yes], [14, 13, 6, 6, Yes], [14, 14, 8, 8, Yes], [14, 15, 10, 10, Yes], [14, 16, 12, 12, Yes], [14, 17, 14, 14, Yes], [14, 18, 16, 16, Yes], [14, 19, 18, 18, Yes], [14, 20, 20, 20, Yes], [14, 21, 22, 22, Yes], [14, 22, 24, 24, Yes], [14, 23, 26, 26, Yes], [15, 0, 30, 30, Yes], [15, 1, 32, 32, Yes], [15, 2, 34, 34, Yes], [15, 3, 36, 36, Yes], [15, 4, 38, 38, Yes], [15, 5, 40, 40, Yes], [15, 6, 42, 42, Yes], [15, 7, 44, 44, Yes], [15, 8, 46, 46, Yes], [15, 9, 0, 0, Yes], [15, 10, 2, 2, Yes], [15, 11, 4, 4, Yes], [15, 12, 6, 6, Yes], [15, 13, 8, 8, Yes], [15, 14, 10, 10, Yes], [15, 15, 12, 12, Yes], [15, 16, 14, 14, Yes], [15, 17, 16, 16, Yes], [15, 18, 18, 18, Yes], [15, 19, 20, 20, Yes], [15, 20, 22, 22, Yes], [15, 21, 24, 24, Yes], [15, 22, 26, 26, Yes], [15, 23, 28, 28, Yes], [16, 0, 32, 32, Yes], [16, 1, 34, 34, Yes], [16, 2, 36, 36, Yes], [16, 3, 38, 38, Yes], [16, 4, 40, 40, Yes], [16, 5, 42, 42, Yes], [16, 6, 44, 44, Yes], [16, 7, 46, 46, Yes], [16, 8, 0, 0, Yes], [16, 9, 2, 2, Yes], [16, 10, 4, 4, Yes], [16, 11, 6, 6, Yes], [16, 12, 8, 8, Yes], [16, 13, 10, 10, Yes], [16, 14, 12, 12, Yes], [16, 15, 14, 14, Yes], [16, 16, 16, 16, Yes], [16, 17, 18, 18, Yes], [16, 18, 20, 20, Yes], [16, 19, 22, 22, Yes], [16, 20, 24, 24, Yes], [16, 21, 26, 26, Yes], [16, 22, 28, 28, Yes], [16, 23, 30, 30, Yes], [17, 0, 34, 34, Yes], [17, 1, 36, 36, Yes], [17, 2, 38, 38, Yes], [17, 3, 40, 40, Yes], [17, 4, 42, 42, Yes], [17, 5, 44, 44, Yes], [17, 6, 46, 46, Yes], [17, 7, 0, 0, Yes], [17, 8, 2, 2, Yes], [17, 9, 4, 4, Yes], [17, 10, 6, 6, Yes], [17, 11, 8, 8, Yes], [17, 12, 10, 10, Yes], [17, 13, 12, 12, Yes], [17, 14, 14, 14, Yes], [17, 15, 16, 16, Yes], [17, 16, 18, 18, Yes], [17, 17, 20, 20, Yes], [17, 18, 22, 22, Yes], [17, 19, 24, 24, Yes], [17, 20, 26, 26, Yes], [17, 21, 28, 28, Yes], [17, 22, 30, 30, Yes], [17, 23, 32, 32, Yes], [18, 0, 36, 36, Yes], [18, 1, 38, 38, Yes], [18, 2, 40, 40, Yes], [18, 3, 42, 42, Yes], [18, 4, 44, 44, Yes], [18, 5, 46, 46, Yes], [18, 6, 0, 0, Yes], [18, 7, 2, 2, Yes], [18, 8, 4, 4, Yes], [18, 9, 6, 6, Yes], [18, 10, 8, 8, Yes], [18, 11, 10, 10, Yes], [18, 12, 12, 12, Yes], [18, 13, 14, 14, Yes], [18, 14, 16, 16, Yes], [18, 15, 18, 18, Yes], [18, 16, 20, 20, Yes], [18, 17, 22, 22, Yes], [18, 18, 24, 24, Yes], [18, 19, 26, 26, Yes], [18, 20, 28, 28, Yes], [18, 21, 30, 30, Yes], [18, 22, 32, 32, Yes], [18, 23, 34, 34, Yes], [19, 0, 38, 38, Yes], [19, 1, 40, 40, Yes], [19, 2, 42, 42, Yes], [19, 3, 44, 44, Yes], [19, 4, 46, 46, Yes], [19, 5, 0, 0, Yes], [19, 6, 2, 2, Yes], [19, 7, 4, 4, Yes], [19, 8, 6, 6, Yes], [19, 9, 8, 8, Yes], [19, 10, 10, 10, Yes], [19, 11, 12, 12, Yes], [19, 12, 14, 14, Yes], [19, 13, 16, 16, Yes], [19, 14, 18, 18, Yes], [19, 15, 20, 20, Yes], [19, 16, 22, 22, Yes], [19, 17, 24, 24, Yes], [19, 18, 26, 26, Yes], [19, 19, 28, 28, Yes], [19, 20, 30, 30, Yes], [19, 21, 32, 32, Yes], [19, 22, 34, 34, Yes], [19, 23, 36, 36, Yes], [20, 0, 40, 40, Yes], [20, 1, 42, 42, Yes], [20, 2, 44, 44, Yes], [20, 3, 46, 46, Yes], [20, 4, 0, 0, Yes], [20, 5, 2, 2, Yes], [20, 6, 4, 4, Yes], [20, 7, 6, 6, Yes], [20, 8, 8, 8, Yes], [20, 9, 10, 10, Yes], [20, 10, 12, 12, Yes], [20, 11, 14, 14, Yes], [20, 12, 16, 16, Yes], [20, 13, 18, 18, Yes], [20, 14, 20, 20, Yes], [20, 15, 22, 22, Yes], [20, 16, 24, 24, Yes], [20, 17, 26, 26, Yes], [20, 18, 28, 28, Yes], [20, 19, 30, 30, Yes], [20, 20, 32, 32, Yes], [20, 21, 34, 34, Yes], [20, 22, 36, 36, Yes], [20, 23, 38, 38, Yes], [21, 0, 42, 42, Yes], [21, 1, 44, 44, Yes], [21, 2, 46, 46, Yes], [21, 3, 0, 0, Yes], [21, 4, 2, 2, Yes], [21, 5, 4, 4, Yes], [21, 6, 6, 6, Yes], [21, 7, 8, 8, Yes], [21, 8, 10, 10, Yes], [21, 9, 12, 12, Yes], [21, 10, 14, 14, Yes], [21, 11, 16, 16, Yes], [21, 12, 18, 18, Yes], [21, 13, 20, 20, Yes], [21, 14, 22, 22, Yes], [21, 15, 24, 24, Yes], [21, 16, 26, 26, Yes], [21, 17, 28, 28, Yes], [21, 18, 30, 30, Yes], [21, 19, 32, 32, Yes], [21, 20, 34, 34, Yes], [21, 21, 36, 36, Yes], [21, 22, 38, 38, Yes], [21, 23, 40, 40, Yes], [22, 0, 44, 44, Yes], [22, 1, 46, 46, Yes], [22, 2, 0, 0, Yes], [22, 3, 2, 2, Yes], [22, 4, 4, 4, Yes], [22, 5, 6, 6, Yes], [22, 6, 8, 8, Yes], [22, 7, 10, 10, Yes], [22, 8, 12, 12, Yes], [22, 9, 14, 14, Yes], [22, 10, 16, 16, Yes], [22, 11, 18, 18, Yes], [22, 12, 20, 20, Yes], [22, 13, 22, 22, Yes], [22, 14, 24, 24, Yes], [22, 15, 26, 26, Yes], [22, 16, 28, 28, Yes], [22, 17, 30, 30, Yes], [22, 18, 32, 32, Yes], [22, 19, 34, 34, Yes], [22, 20, 36, 36, Yes], [22, 21, 38, 38, Yes], [22, 22, 40, 40, Yes], [22, 23, 42, 42, Yes], [23, 0, 46, 46, Yes], [23, 1, 0, 0, Yes], [23, 2, 2, 2, Yes], [23, 3, 4, 4, Yes], [23, 4, 6, 6, Yes], [23, 5, 8, 8, Yes], [23, 6, 10, 10, Yes], [23, 7, 12, 12, Yes], [23, 8, 14, 14, Yes], [23, 9, 16, 16, Yes], [23, 10, 18, 18, Yes], [23, 11, 20, 20, Yes], [23, 12, 22, 22, Yes], [23, 13, 24, 24, Yes], [23, 14, 26, 26, Yes], [23, 15, 28, 28, Yes], [23, 16, 30, 30, Yes], [23, 17, 32, 32, Yes], [23, 18, 34, 34, Yes], [23, 19, 36, 36, Yes], [23, 20, 38, 38, Yes], [23, 21, 40, 40, Yes], [23, 22, 42, 42, Yes], [23, 23, 44, 44, Yes]], Yes, null, [0], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], Yes, Yes] Conditions: none Audit every homomorphism pair and the full target-identity preimage/image.Qualification allows a bad map and reports actual properties:homFalse can coexist with normal kernel or subgroup image.An empty kernel is not replaced by an identity fallback. Step identifiers and dependenciesRule: Iso1_ALL_product_laws_ACTUAL_kernel_image_properties_before_theorem · Uses: G001, S007 S009 · impliesBefore source roles = [[0], Yes, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46]] After derived = [[[0], [1], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [2], [20], [21], [22], [23], [3], [4], [5], [6], [7], [8], [9]], [0, 1, 12, 17, 18, 19, 20, 21, 22, 23, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16], [[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, 12, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 17, 14, 15, 16, 0, 18, 19, 20, 21, 22, 23, 2], [2, 3, 13, 14, 15, 16, 0, 1, 12, 17, 18, 19, 4, 20, 21, 22, 23, 5, 6, 7, 8, 9, 10, 11], [3, 4, 14, 15, 16, 0, 1, 12, 17, 18, 19, 20, 5, 21, 22, 23, 2, 6, 7, 8, 9, 10, 11, 13], [4, 5, 15, 16, 0, 1, 12, 17, 18, 19, 20, 21, 6, 22, 23, 2, 3, 7, 8, 9, 10, 11, 13, 14], [5, 6, 16, 0, 1, 12, 17, 18, 19, 20, 21, 22, 7, 23, 2, 3, 4, 8, 9, 10, 11, 13, 14, 15], [6, 7, 0, 1, 12, 17, 18, 19, 20, 21, 22, 23, 8, 2, 3, 4, 5, 9, 10, 11, 13, 14, 15, 16], [7, 8, 1, 12, 17, 18, 19, 20, 21, 22, 23, 2, 9, 3, 4, 5, 6, 10, 11, 13, 14, 15, 16, 0], [8, 9, 12, 17, 18, 19, 20, 21, 22, 23, 2, 3, 10, 4, 5, 6, 7, 11, 13, 14, 15, 16, 0, 1], [9, 10, 17, 18, 19, 20, 21, 22, 23, 2, 3, 4, 11, 5, 6, 7, 8, 13, 14, 15, 16, 0, 1, 12], [10, 11, 18, 19, 20, 21, 22, 23, 2, 3, 4, 5, 13, 6, 7, 8, 9, 14, 15, 16, 0, 1, 12, 17], [11, 13, 19, 20, 21, 22, 23, 2, 3, 4, 5, 6, 14, 7, 8, 9, 10, 15, 16, 0, 1, 12, 17, 18], [12, 17, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 18, 15, 16, 0, 1, 19, 20, 21, 22, 23, 2, 3], [13, 14, 20, 21, 22, 23, 2, 3, 4, 5, 6, 7, 15, 8, 9, 10, 11, 16, 0, 1, 12, 17, 18, 19], [14, 15, 21, 22, 23, 2, 3, 4, 5, 6, 7, 8, 16, 9, 10, 11, 13, 0, 1, 12, 17, 18, 19, 20], [15, 16, 22, 23, 2, 3, 4, 5, 6, 7, 8, 9, 0, 10, 11, 13, 14, 1, 12, 17, 18, 19, 20, 21], [16, 0, 23, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 11, 13, 14, 15, 12, 17, 18, 19, 20, 21, 22], [17, 18, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 19, 16, 0, 1, 12, 20, 21, 22, 23, 2, 3, 4], [18, 19, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 20, 0, 1, 12, 17, 21, 22, 23, 2, 3, 4, 5], [19, 20, 7, 8, 9, 10, 11, 13, 14, 15, 16, 0, 21, 1, 12, 17, 18, 22, 23, 2, 3, 4, 5, 6], [20, 21, 8, 9, 10, 11, 13, 14, 15, 16, 0, 1, 22, 12, 17, 18, 19, 23, 2, 3, 4, 5, 6, 7], [21, 22, 9, 10, 11, 13, 14, 15, 16, 0, 1, 12, 23, 17, 18, 19, 20, 2, 3, 4, 5, 6, 7, 8], [22, 23, 10, 11, 13, 14, 15, 16, 0, 1, 12, 17, 2, 18, 19, 20, 21, 3, 4, 5, 6, 7, 8, 9], [23, 2, 11, 13, 14, 15, 16, 0, 1, 12, 17, 18, 3, 19, 20, 21, 22, 4, 5, 6, 7, 8, 9, 10]], [[0, 0, [0], Yes], [0, 1, [1], Yes], [0, 2, [2], Yes], [0, 3, [3], Yes], [0, 4, [4], Yes], [0, 5, [5], Yes], [0, 6, [6], Yes], [0, 7, [7], Yes], [0, 8, [8], Yes], [0, 9, [9], Yes], [0, 10, [10], Yes], [0, 11, [11], Yes], [0, 12, [12], Yes], [0, 13, [13], Yes], [0, 14, [14], Yes], [0, 15, [15], Yes], [0, 16, [16], Yes], [0, 17, [17], Yes], [0, 18, [18], Yes], [0, 19, [19], Yes], [0, 20, [20], Yes], [0, 21, [21], Yes], [0, 22, [22], Yes], [0, 23, [23], Yes], [1, 0, [1], Yes], [1, 1, [12], Yes], [1, 2, [3], Yes], [1, 3, [4], Yes], [1, 4, [5], Yes], [1, 5, [6], Yes], [1, 6, [7], Yes], [1, 7, [8], Yes], [1, 8, [9], Yes], [1, 9, [10], Yes], [1, 10, [11], Yes], [1, 11, [13], Yes], [1, 12, [17], Yes], [1, 13, [14], Yes], [1, 14, [15], Yes], [1, 15, [16], Yes], [1, 16, [0], Yes], [1, 17, [18], Yes], [1, 18, [19], Yes], [1, 19, [20], Yes], [1, 20, [21], Yes], [1, 21, [22], Yes], [1, 22, [23], Yes], [1, 23, [2], Yes], [2, 0, [2], Yes], [2, 1, [3], Yes], [2, 2, [13], Yes], [2, 3, [14], Yes], [2, 4, [15], Yes], [2, 5, [16], Yes], [2, 6, [0], Yes], [2, 7, [1], Yes], [2, 8, [12], Yes], [2, 9, [17], Yes], [2, 10, [18], Yes], [2, 11, [19], Yes], [2, 12, [4], Yes], [2, 13, [20], Yes], [2, 14, [21], Yes], [2, 15, [22], Yes], [2, 16, [23], Yes], [2, 17, [5], Yes], [2, 18, [6], Yes], [2, 19, [7], Yes], [2, 20, [8], Yes], [2, 21, [9], Yes], [2, 22, [10], Yes], [2, 23, [11], Yes], [3, 0, [3], Yes], [3, 1, [4], Yes], [3, 2, [14], Yes], [3, 3, [15], Yes], [3, 4, [16], Yes], [3, 5, [0], Yes], [3, 6, [1], Yes], [3, 7, [12], Yes], [3, 8, [17], Yes], [3, 9, [18], Yes], [3, 10, [19], Yes], [3, 11, [20], Yes], [3, 12, [5], Yes], [3, 13, [21], Yes], [3, 14, [22], Yes], [3, 15, [23], Yes], [3, 16, [2], Yes], [3, 17, [6], Yes], [3, 18, [7], Yes], [3, 19, [8], Yes], [3, 20, [9], Yes], [3, 21, [10], Yes], [3, 22, [11], Yes], [3, 23, [13], Yes], [4, 0, [4], Yes], [4, 1, [5], Yes], [4, 2, [15], Yes], [4, 3, [16], Yes], [4, 4, [0], Yes], [4, 5, [1], Yes], [4, 6, [12], Yes], [4, 7, [17], Yes], [4, 8, [18], Yes], [4, 9, [19], Yes], [4, 10, [20], Yes], [4, 11, [21], Yes], [4, 12, [6], Yes], [4, 13, [22], Yes], [4, 14, [23], Yes], [4, 15, [2], Yes], [4, 16, [3], Yes], [4, 17, [7], Yes], [4, 18, [8], Yes], [4, 19, [9], Yes], [4, 20, [10], Yes], [4, 21, [11], Yes], [4, 22, [13], Yes], [4, 23, [14], Yes], [5, 0, [5], Yes], [5, 1, [6], Yes], [5, 2, [16], Yes], [5, 3, [0], Yes], [5, 4, [1], Yes], [5, 5, [12], Yes], [5, 6, [17], Yes], [5, 7, [18], Yes], [5, 8, [19], Yes], [5, 9, [20], Yes], [5, 10, [21], Yes], [5, 11, [22], Yes], [5, 12, [7], Yes], [5, 13, [23], Yes], [5, 14, [2], Yes], [5, 15, [3], Yes], [5, 16, [4], Yes], [5, 17, [8], Yes], [5, 18, [9], Yes], [5, 19, [10], Yes], [5, 20, [11], Yes], [5, 21, [13], Yes], [5, 22, [14], Yes], [5, 23, [15], Yes], [6, 0, [6], Yes], [6, 1, [7], Yes], [6, 2, [0], Yes], [6, 3, [1], Yes], [6, 4, [12], Yes], [6, 5, [17], Yes], [6, 6, [18], Yes], [6, 7, [19], Yes], [6, 8, [20], Yes], [6, 9, [21], Yes], [6, 10, [22], Yes], [6, 11, [23], Yes], [6, 12, [8], Yes], [6, 13, [2], Yes], [6, 14, [3], Yes], [6, 15, [4], Yes], [6, 16, [5], Yes], [6, 17, [9], Yes], [6, 18, [10], Yes], [6, 19, [11], Yes], [6, 20, [13], Yes], [6, 21, [14], Yes], [6, 22, [15], Yes], [6, 23, [16], Yes], [7, 0, [7], Yes], [7, 1, [8], Yes], [7, 2, [1], Yes], [7, 3, [12], Yes], [7, 4, [17], Yes], [7, 5, [18], Yes], [7, 6, [19], Yes], [7, 7, [20], Yes], [7, 8, [21], Yes], [7, 9, [22], Yes], [7, 10, [23], Yes], [7, 11, [2], Yes], [7, 12, [9], Yes], [7, 13, [3], Yes], [7, 14, [4], Yes], [7, 15, [5], Yes], [7, 16, [6], Yes], [7, 17, [10], Yes], [7, 18, [11], Yes], [7, 19, [13], Yes], [7, 20, [14], Yes], [7, 21, [15], Yes], [7, 22, [16], Yes], [7, 23, [0], Yes], [8, 0, [8], Yes], [8, 1, [9], Yes], [8, 2, [12], Yes], [8, 3, [17], Yes], [8, 4, [18], Yes], [8, 5, [19], Yes], [8, 6, [20], Yes], [8, 7, [21], Yes], [8, 8, [22], Yes], [8, 9, [23], Yes], [8, 10, [2], Yes], [8, 11, [3], Yes], [8, 12, [10], Yes], [8, 13, [4], Yes], [8, 14, [5], Yes], [8, 15, [6], Yes], [8, 16, [7], Yes], [8, 17, [11], Yes], [8, 18, [13], Yes], [8, 19, [14], Yes], [8, 20, [15], Yes], [8, 21, [16], Yes], [8, 22, [0], Yes], [8, 23, [1], Yes], [9, 0, [9], Yes], [9, 1, [10], Yes], [9, 2, [17], Yes], [9, 3, [18], Yes], [9, 4, [19], Yes], [9, 5, [20], Yes], [9, 6, [21], Yes], [9, 7, [22], Yes], [9, 8, [23], Yes], [9, 9, [2], Yes], [9, 10, [3], Yes], [9, 11, [4], Yes], [9, 12, [11], Yes], [9, 13, [5], Yes], [9, 14, [6], Yes], [9, 15, [7], Yes], [9, 16, [8], Yes], [9, 17, [13], Yes], [9, 18, [14], Yes], [9, 19, [15], Yes], [9, 20, [16], Yes], [9, 21, [0], Yes], [9, 22, [1], Yes], [9, 23, [12], Yes], [10, 0, [10], Yes], [10, 1, [11], Yes], [10, 2, [18], Yes], [10, 3, [19], Yes], [10, 4, [20], Yes], [10, 5, [21], Yes], [10, 6, [22], Yes], [10, 7, [23], Yes], [10, 8, [2], Yes], [10, 9, [3], Yes], [10, 10, [4], Yes], [10, 11, [5], Yes], [10, 12, [13], Yes], [10, 13, [6], Yes], [10, 14, [7], Yes], [10, 15, [8], Yes], [10, 16, [9], Yes], [10, 17, [14], Yes], [10, 18, [15], Yes], [10, 19, [16], Yes], [10, 20, [0], Yes], [10, 21, [1], Yes], [10, 22, [12], Yes], [10, 23, [17], Yes], [11, 0, [11], Yes], [11, 1, [13], Yes], [11, 2, [19], Yes], [11, 3, [20], Yes], [11, 4, [21], Yes], [11, 5, [22], Yes], [11, 6, [23], Yes], [11, 7, [2], Yes], [11, 8, [3], Yes], [11, 9, [4], Yes], [11, 10, [5], Yes], [11, 11, [6], Yes], [11, 12, [14], Yes], [11, 13, [7], Yes], [11, 14, [8], Yes], [11, 15, [9], Yes], [11, 16, [10], Yes], [11, 17, [15], Yes], [11, 18, [16], Yes], [11, 19, [0], Yes], [11, 20, [1], Yes], [11, 21, [12], Yes], [11, 22, [17], Yes], [11, 23, [18], Yes], [12, 0, [12], Yes], [12, 1, [17], Yes], [12, 2, [4], Yes], [12, 3, [5], Yes], [12, 4, [6], Yes], [12, 5, [7], Yes], [12, 6, [8], Yes], [12, 7, [9], Yes], [12, 8, [10], Yes], [12, 9, [11], Yes], [12, 10, [13], Yes], [12, 11, [14], Yes], [12, 12, [18], Yes], [12, 13, [15], Yes], [12, 14, [16], Yes], [12, 15, [0], Yes], [12, 16, [1], Yes], [12, 17, [19], Yes], [12, 18, [20], Yes], [12, 19, [21], Yes], [12, 20, [22], Yes], [12, 21, [23], Yes], [12, 22, [2], Yes], [12, 23, [3], Yes], [13, 0, [13], Yes], [13, 1, [14], Yes], [13, 2, [20], Yes], [13, 3, [21], Yes], [13, 4, [22], Yes], [13, 5, [23], Yes], [13, 6, [2], Yes], [13, 7, [3], Yes], [13, 8, [4], Yes], [13, 9, [5], Yes], [13, 10, [6], Yes], [13, 11, [7], Yes], [13, 12, [15], Yes], [13, 13, [8], Yes], [13, 14, [9], Yes], [13, 15, [10], Yes], [13, 16, [11], Yes], [13, 17, [16], Yes], [13, 18, [0], Yes], [13, 19, [1], Yes], [13, 20, [12], Yes], [13, 21, [17], Yes], [13, 22, [18], Yes], [13, 23, [19], Yes], [14, 0, [14], Yes], [14, 1, [15], Yes], [14, 2, [21], Yes], [14, 3, [22], Yes], [14, 4, [23], Yes], [14, 5, [2], Yes], [14, 6, [3], Yes], [14, 7, [4], Yes], [14, 8, [5], Yes], [14, 9, [6], Yes], [14, 10, [7], Yes], [14, 11, [8], Yes], [14, 12, [16], Yes], [14, 13, [9], Yes], [14, 14, [10], Yes], [14, 15, [11], Yes], [14, 16, [13], Yes], [14, 17, [0], Yes], [14, 18, [1], Yes], [14, 19, [12], Yes], [14, 20, [17], Yes], [14, 21, [18], Yes], [14, 22, [19], Yes], [14, 23, [20], Yes], [15, 0, [15], Yes], [15, 1, [16], Yes], [15, 2, [22], Yes], [15, 3, [23], Yes], [15, 4, [2], Yes], [15, 5, [3], Yes], [15, 6, [4], Yes], [15, 7, [5], Yes], [15, 8, [6], Yes], [15, 9, [7], Yes], [15, 10, [8], Yes], [15, 11, [9], Yes], [15, 12, [0], Yes], [15, 13, [10], Yes], [15, 14, [11], Yes], [15, 15, [13], Yes], [15, 16, [14], Yes], [15, 17, [1], Yes], [15, 18, [12], Yes], [15, 19, [17], Yes], [15, 20, [18], Yes], [15, 21, [19], Yes], [15, 22, [20], Yes], [15, 23, [21], Yes], [16, 0, [16], Yes], [16, 1, [0], Yes], [16, 2, [23], Yes], [16, 3, [2], Yes], [16, 4, [3], Yes], [16, 5, [4], Yes], [16, 6, [5], Yes], [16, 7, [6], Yes], [16, 8, [7], Yes], [16, 9, [8], Yes], [16, 10, [9], Yes], [16, 11, [10], Yes], [16, 12, [1], Yes], [16, 13, [11], Yes], [16, 14, [13], Yes], [16, 15, [14], Yes], [16, 16, [15], Yes], [16, 17, [12], Yes], [16, 18, [17], Yes], [16, 19, [18], Yes], [16, 20, [19], Yes], [16, 21, [20], Yes], [16, 22, [21], Yes], [16, 23, [22], Yes], [17, 0, [17], Yes], [17, 1, [18], Yes], [17, 2, [5], Yes], [17, 3, [6], Yes], [17, 4, [7], Yes], [17, 5, [8], Yes], [17, 6, [9], Yes], [17, 7, [10], Yes], [17, 8, [11], Yes], [17, 9, [13], Yes], [17, 10, [14], Yes], [17, 11, [15], Yes], [17, 12, [19], Yes], [17, 13, [16], Yes], [17, 14, [0], Yes], [17, 15, [1], Yes], [17, 16, [12], Yes], [17, 17, [20], Yes], [17, 18, [21], Yes], [17, 19, [22], Yes], [17, 20, [23], Yes], [17, 21, [2], Yes], [17, 22, [3], Yes], [17, 23, [4], Yes], [18, 0, [18], Yes], [18, 1, [19], Yes], [18, 2, [6], Yes], [18, 3, [7], Yes], [18, 4, [8], Yes], [18, 5, [9], Yes], [18, 6, [10], Yes], [18, 7, [11], Yes], [18, 8, [13], Yes], [18, 9, [14], Yes], [18, 10, [15], Yes], [18, 11, [16], Yes], [18, 12, [20], Yes], [18, 13, [0], Yes], [18, 14, [1], Yes], [18, 15, [12], Yes], [18, 16, [17], Yes], [18, 17, [21], Yes], [18, 18, [22], Yes], [18, 19, [23], Yes], [18, 20, [2], Yes], [18, 21, [3], Yes], [18, 22, [4], Yes], [18, 23, [5], Yes], [19, 0, [19], Yes], [19, 1, [20], Yes], [19, 2, [7], Yes], [19, 3, [8], Yes], [19, 4, [9], Yes], [19, 5, [10], Yes], [19, 6, [11], Yes], [19, 7, [13], Yes], [19, 8, [14], Yes], [19, 9, [15], Yes], [19, 10, [16], Yes], [19, 11, [0], Yes], [19, 12, [21], Yes], [19, 13, [1], Yes], [19, 14, [12], Yes], [19, 15, [17], Yes], [19, 16, [18], Yes], [19, 17, [22], Yes], [19, 18, [23], Yes], [19, 19, [2], Yes], [19, 20, [3], Yes], [19, 21, [4], Yes], [19, 22, [5], Yes], [19, 23, [6], Yes], [20, 0, [20], Yes], [20, 1, [21], Yes], [20, 2, [8], Yes], [20, 3, [9], Yes], [20, 4, [10], Yes], [20, 5, [11], Yes], [20, 6, [13], Yes], [20, 7, [14], Yes], [20, 8, [15], Yes], [20, 9, [16], Yes], [20, 10, [0], Yes], [20, 11, [1], Yes], [20, 12, [22], Yes], [20, 13, [12], Yes], [20, 14, [17], Yes], [20, 15, [18], Yes], [20, 16, [19], Yes], [20, 17, [23], Yes], [20, 18, [2], Yes], [20, 19, [3], Yes], [20, 20, [4], Yes], [20, 21, [5], Yes], [20, 22, [6], Yes], [20, 23, [7], Yes], [21, 0, [21], Yes], [21, 1, [22], Yes], [21, 2, [9], Yes], [21, 3, [10], Yes], [21, 4, [11], Yes], [21, 5, [13], Yes], [21, 6, [14], Yes], [21, 7, [15], Yes], [21, 8, [16], Yes], [21, 9, [0], Yes], [21, 10, [1], Yes], [21, 11, [12], Yes], [21, 12, [23], Yes], [21, 13, [17], Yes], [21, 14, [18], Yes], [21, 15, [19], Yes], [21, 16, [20], Yes], [21, 17, [2], Yes], [21, 18, [3], Yes], [21, 19, [4], Yes], [21, 20, [5], Yes], [21, 21, [6], Yes], [21, 22, [7], Yes], [21, 23, [8], Yes], [22, 0, [22], Yes], [22, 1, [23], Yes], [22, 2, [10], Yes], [22, 3, [11], Yes], [22, 4, [13], Yes], [22, 5, [14], Yes], [22, 6, [15], Yes], [22, 7, [16], Yes], [22, 8, [0], Yes], [22, 9, [1], Yes], [22, 10, [12], Yes], [22, 11, [17], Yes], [22, 12, [2], Yes], [22, 13, [18], Yes], [22, 14, [19], Yes], [22, 15, [20], Yes], [22, 16, [21], Yes], [22, 17, [3], Yes], [22, 18, [4], Yes], [22, 19, [5], Yes], [22, 20, [6], Yes], [22, 21, [7], Yes], [22, 22, [8], Yes], [22, 23, [9], Yes], [23, 0, [23], Yes], [23, 1, [2], Yes], [23, 2, [11], Yes], [23, 3, [13], Yes], [23, 4, [14], Yes], [23, 5, [15], Yes], [23, 6, [16], Yes], [23, 7, [0], Yes], [23, 8, [1], Yes], [23, 9, [12], Yes], [23, 10, [17], Yes], [23, 11, [18], Yes], [23, 12, [3], Yes], [23, 13, [19], Yes], [23, 14, [20], Yes], [23, 15, [21], Yes], [23, 16, [22], Yes], [23, 17, [4], Yes], [23, 18, [5], Yes], [23, 19, [6], Yes], [23, 20, [7], Yes], [23, 21, [8], Yes], [23, 22, [9], Yes], [23, 23, [10], Yes]], [[0, [0], [0], Yes], [1, [1], [2], Yes], [2, [10], [20], Yes], [3, [11], [22], Yes], [4, [12], [24], Yes], [5, [13], [26], Yes], [6, [14], [28], Yes], [7, [15], [30], Yes], [8, [16], [32], Yes], [9, [17], [34], Yes], [10, [18], [36], Yes], [11, [19], [38], Yes], [12, [2], [4], Yes], [13, [20], [40], Yes], [14, [21], [42], Yes], [15, [22], [44], Yes], [16, [23], [46], Yes], [17, [3], [6], Yes], [18, [4], [8], Yes], [19, [5], [10], Yes], [20, [6], [12], Yes], [21, [7], [14], Yes], [22, [8], [16], Yes], [23, [9], [18], Yes]], Yes] Conditions: none Kernel normality gives(ah)(bk)=ab(b^-1hb)k in abK;check every representative product agrees on its quotient cell.Every f(g) in a coset must also agree before defining bar_f.The carrier consists of coset SETS,not least representatives. Step identifiers and dependenciesRule: Iso1_full_LEFT_coset_quotient_ALL_products_and_representative_images · Uses: G001, S008 S010 · impliesBefore source roles = [[[0], [1], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [2], [20], [21], [22], [23], [3], [4], [5], [6], [7], [8], [9]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], [0, 2, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 4, 40, 42, 44, 46, 6, 8, 10, 12, 14, 16, 18]] After derived = [[[0, 0, 0, 0, Yes], [0, 1, 2, 2, Yes], [0, 2, 20, 20, Yes], [0, 3, 22, 22, Yes], [0, 4, 24, 24, Yes], [0, 5, 26, 26, Yes], [0, 6, 28, 28, Yes], [0, 7, 30, 30, Yes], [0, 8, 32, 32, Yes], [0, 9, 34, 34, Yes], [0, 10, 36, 36, Yes], [0, 11, 38, 38, Yes], [0, 12, 4, 4, Yes], [0, 13, 40, 40, Yes], [0, 14, 42, 42, Yes], [0, 15, 44, 44, Yes], [0, 16, 46, 46, Yes], [0, 17, 6, 6, Yes], [0, 18, 8, 8, Yes], [0, 19, 10, 10, Yes], [0, 20, 12, 12, Yes], [0, 21, 14, 14, Yes], [0, 22, 16, 16, Yes], [0, 23, 18, 18, Yes], [1, 0, 2, 2, Yes], [1, 1, 4, 4, Yes], [1, 2, 22, 22, Yes], [1, 3, 24, 24, Yes], [1, 4, 26, 26, Yes], [1, 5, 28, 28, Yes], [1, 6, 30, 30, Yes], [1, 7, 32, 32, Yes], [1, 8, 34, 34, Yes], [1, 9, 36, 36, Yes], [1, 10, 38, 38, Yes], [1, 11, 40, 40, Yes], [1, 12, 6, 6, Yes], [1, 13, 42, 42, Yes], [1, 14, 44, 44, Yes], [1, 15, 46, 46, Yes], [1, 16, 0, 0, Yes], [1, 17, 8, 8, Yes], [1, 18, 10, 10, Yes], [1, 19, 12, 12, Yes], [1, 20, 14, 14, Yes], [1, 21, 16, 16, Yes], [1, 22, 18, 18, Yes], [1, 23, 20, 20, Yes], [2, 0, 20, 20, Yes], [2, 1, 22, 22, Yes], [2, 2, 40, 40, Yes], [2, 3, 42, 42, Yes], [2, 4, 44, 44, Yes], [2, 5, 46, 46, Yes], [2, 6, 0, 0, Yes], [2, 7, 2, 2, Yes], [2, 8, 4, 4, Yes], [2, 9, 6, 6, Yes], [2, 10, 8, 8, Yes], [2, 11, 10, 10, Yes], [2, 12, 24, 24, Yes], [2, 13, 12, 12, Yes], [2, 14, 14, 14, Yes], [2, 15, 16, 16, Yes], [2, 16, 18, 18, Yes], [2, 17, 26, 26, Yes], [2, 18, 28, 28, Yes], [2, 19, 30, 30, Yes], [2, 20, 32, 32, Yes], [2, 21, 34, 34, Yes], [2, 22, 36, 36, Yes], [2, 23, 38, 38, Yes], [3, 0, 22, 22, Yes], [3, 1, 24, 24, Yes], [3, 2, 42, 42, Yes], [3, 3, 44, 44, Yes], [3, 4, 46, 46, Yes], [3, 5, 0, 0, Yes], [3, 6, 2, 2, Yes], [3, 7, 4, 4, Yes], [3, 8, 6, 6, Yes], [3, 9, 8, 8, Yes], [3, 10, 10, 10, Yes], [3, 11, 12, 12, Yes], [3, 12, 26, 26, Yes], [3, 13, 14, 14, Yes], [3, 14, 16, 16, Yes], [3, 15, 18, 18, Yes], [3, 16, 20, 20, Yes], [3, 17, 28, 28, Yes], [3, 18, 30, 30, Yes], [3, 19, 32, 32, Yes], [3, 20, 34, 34, Yes], [3, 21, 36, 36, Yes], [3, 22, 38, 38, Yes], [3, 23, 40, 40, Yes], [4, 0, 24, 24, Yes], [4, 1, 26, 26, Yes], [4, 2, 44, 44, Yes], [4, 3, 46, 46, Yes], [4, 4, 0, 0, Yes], [4, 5, 2, 2, Yes], [4, 6, 4, 4, Yes], [4, 7, 6, 6, Yes], [4, 8, 8, 8, Yes], [4, 9, 10, 10, Yes], [4, 10, 12, 12, Yes], [4, 11, 14, 14, Yes], [4, 12, 28, 28, Yes], [4, 13, 16, 16, Yes], [4, 14, 18, 18, Yes], [4, 15, 20, 20, Yes], [4, 16, 22, 22, Yes], [4, 17, 30, 30, Yes], [4, 18, 32, 32, Yes], [4, 19, 34, 34, Yes], [4, 20, 36, 36, Yes], [4, 21, 38, 38, Yes], [4, 22, 40, 40, Yes], [4, 23, 42, 42, Yes], [5, 0, 26, 26, Yes], [5, 1, 28, 28, Yes], [5, 2, 46, 46, Yes], [5, 3, 0, 0, Yes], [5, 4, 2, 2, Yes], [5, 5, 4, 4, Yes], [5, 6, 6, 6, Yes], [5, 7, 8, 8, Yes], [5, 8, 10, 10, Yes], [5, 9, 12, 12, Yes], [5, 10, 14, 14, Yes], [5, 11, 16, 16, Yes], [5, 12, 30, 30, Yes], [5, 13, 18, 18, Yes], [5, 14, 20, 20, Yes], [5, 15, 22, 22, Yes], [5, 16, 24, 24, Yes], [5, 17, 32, 32, Yes], [5, 18, 34, 34, Yes], [5, 19, 36, 36, Yes], [5, 20, 38, 38, Yes], [5, 21, 40, 40, Yes], [5, 22, 42, 42, Yes], [5, 23, 44, 44, Yes], [6, 0, 28, 28, Yes], [6, 1, 30, 30, Yes], [6, 2, 0, 0, Yes], [6, 3, 2, 2, Yes], [6, 4, 4, 4, Yes], [6, 5, 6, 6, Yes], [6, 6, 8, 8, Yes], [6, 7, 10, 10, Yes], [6, 8, 12, 12, Yes], [6, 9, 14, 14, Yes], [6, 10, 16, 16, Yes], [6, 11, 18, 18, Yes], [6, 12, 32, 32, Yes], [6, 13, 20, 20, Yes], [6, 14, 22, 22, Yes], [6, 15, 24, 24, Yes], [6, 16, 26, 26, Yes], [6, 17, 34, 34, Yes], [6, 18, 36, 36, Yes], [6, 19, 38, 38, Yes], [6, 20, 40, 40, Yes], [6, 21, 42, 42, Yes], [6, 22, 44, 44, Yes], [6, 23, 46, 46, Yes], [7, 0, 30, 30, Yes], [7, 1, 32, 32, Yes], [7, 2, 2, 2, Yes], [7, 3, 4, 4, Yes], [7, 4, 6, 6, Yes], [7, 5, 8, 8, Yes], [7, 6, 10, 10, Yes], [7, 7, 12, 12, Yes], [7, 8, 14, 14, Yes], [7, 9, 16, 16, Yes], [7, 10, 18, 18, Yes], [7, 11, 20, 20, Yes], [7, 12, 34, 34, Yes], [7, 13, 22, 22, Yes], [7, 14, 24, 24, Yes], [7, 15, 26, 26, Yes], [7, 16, 28, 28, Yes], [7, 17, 36, 36, Yes], [7, 18, 38, 38, Yes], [7, 19, 40, 40, Yes], [7, 20, 42, 42, Yes], [7, 21, 44, 44, Yes], [7, 22, 46, 46, Yes], [7, 23, 0, 0, Yes], [8, 0, 32, 32, Yes], [8, 1, 34, 34, Yes], [8, 2, 4, 4, Yes], [8, 3, 6, 6, Yes], [8, 4, 8, 8, Yes], [8, 5, 10, 10, Yes], [8, 6, 12, 12, Yes], [8, 7, 14, 14, Yes], [8, 8, 16, 16, Yes], [8, 9, 18, 18, Yes], [8, 10, 20, 20, Yes], [8, 11, 22, 22, Yes], [8, 12, 36, 36, Yes], [8, 13, 24, 24, Yes], [8, 14, 26, 26, Yes], [8, 15, 28, 28, Yes], [8, 16, 30, 30, Yes], [8, 17, 38, 38, Yes], [8, 18, 40, 40, Yes], [8, 19, 42, 42, Yes], [8, 20, 44, 44, Yes], [8, 21, 46, 46, Yes], [8, 22, 0, 0, Yes], [8, 23, 2, 2, Yes], [9, 0, 34, 34, Yes], [9, 1, 36, 36, Yes], [9, 2, 6, 6, Yes], [9, 3, 8, 8, Yes], [9, 4, 10, 10, Yes], [9, 5, 12, 12, Yes], [9, 6, 14, 14, Yes], [9, 7, 16, 16, Yes], [9, 8, 18, 18, Yes], [9, 9, 20, 20, Yes], [9, 10, 22, 22, Yes], [9, 11, 24, 24, Yes], [9, 12, 38, 38, Yes], [9, 13, 26, 26, Yes], [9, 14, 28, 28, Yes], [9, 15, 30, 30, Yes], [9, 16, 32, 32, Yes], [9, 17, 40, 40, Yes], [9, 18, 42, 42, Yes], [9, 19, 44, 44, Yes], [9, 20, 46, 46, Yes], [9, 21, 0, 0, Yes], [9, 22, 2, 2, Yes], [9, 23, 4, 4, Yes], [10, 0, 36, 36, Yes], [10, 1, 38, 38, Yes], [10, 2, 8, 8, Yes], [10, 3, 10, 10, Yes], [10, 4, 12, 12, Yes], [10, 5, 14, 14, Yes], [10, 6, 16, 16, Yes], [10, 7, 18, 18, Yes], [10, 8, 20, 20, Yes], [10, 9, 22, 22, Yes], [10, 10, 24, 24, Yes], [10, 11, 26, 26, Yes], [10, 12, 40, 40, Yes], [10, 13, 28, 28, Yes], [10, 14, 30, 30, Yes], [10, 15, 32, 32, Yes], [10, 16, 34, 34, Yes], [10, 17, 42, 42, Yes], [10, 18, 44, 44, Yes], [10, 19, 46, 46, Yes], [10, 20, 0, 0, Yes], [10, 21, 2, 2, Yes], [10, 22, 4, 4, Yes], [10, 23, 6, 6, Yes], [11, 0, 38, 38, Yes], [11, 1, 40, 40, Yes], [11, 2, 10, 10, Yes], [11, 3, 12, 12, Yes], [11, 4, 14, 14, Yes], [11, 5, 16, 16, Yes], [11, 6, 18, 18, Yes], [11, 7, 20, 20, Yes], [11, 8, 22, 22, Yes], [11, 9, 24, 24, Yes], [11, 10, 26, 26, Yes], [11, 11, 28, 28, Yes], [11, 12, 42, 42, Yes], [11, 13, 30, 30, Yes], [11, 14, 32, 32, Yes], [11, 15, 34, 34, Yes], [11, 16, 36, 36, Yes], [11, 17, 44, 44, Yes], [11, 18, 46, 46, Yes], [11, 19, 0, 0, Yes], [11, 20, 2, 2, Yes], [11, 21, 4, 4, Yes], [11, 22, 6, 6, Yes], [11, 23, 8, 8, Yes], [12, 0, 4, 4, Yes], [12, 1, 6, 6, Yes], [12, 2, 24, 24, Yes], [12, 3, 26, 26, Yes], [12, 4, 28, 28, Yes], [12, 5, 30, 30, Yes], [12, 6, 32, 32, Yes], [12, 7, 34, 34, Yes], [12, 8, 36, 36, Yes], [12, 9, 38, 38, Yes], [12, 10, 40, 40, Yes], [12, 11, 42, 42, Yes], [12, 12, 8, 8, Yes], [12, 13, 44, 44, Yes], [12, 14, 46, 46, Yes], [12, 15, 0, 0, Yes], [12, 16, 2, 2, Yes], [12, 17, 10, 10, Yes], [12, 18, 12, 12, Yes], [12, 19, 14, 14, Yes], [12, 20, 16, 16, Yes], [12, 21, 18, 18, Yes], [12, 22, 20, 20, Yes], [12, 23, 22, 22, Yes], [13, 0, 40, 40, Yes], [13, 1, 42, 42, Yes], [13, 2, 12, 12, Yes], [13, 3, 14, 14, Yes], [13, 4, 16, 16, Yes], [13, 5, 18, 18, Yes], [13, 6, 20, 20, Yes], [13, 7, 22, 22, Yes], [13, 8, 24, 24, Yes], [13, 9, 26, 26, Yes], [13, 10, 28, 28, Yes], [13, 11, 30, 30, Yes], [13, 12, 44, 44, Yes], [13, 13, 32, 32, Yes], [13, 14, 34, 34, Yes], [13, 15, 36, 36, Yes], [13, 16, 38, 38, Yes], [13, 17, 46, 46, Yes], [13, 18, 0, 0, Yes], [13, 19, 2, 2, Yes], [13, 20, 4, 4, Yes], [13, 21, 6, 6, Yes], [13, 22, 8, 8, Yes], [13, 23, 10, 10, Yes], [14, 0, 42, 42, Yes], [14, 1, 44, 44, Yes], [14, 2, 14, 14, Yes], [14, 3, 16, 16, Yes], [14, 4, 18, 18, Yes], [14, 5, 20, 20, Yes], [14, 6, 22, 22, Yes], [14, 7, 24, 24, Yes], [14, 8, 26, 26, Yes], [14, 9, 28, 28, Yes], [14, 10, 30, 30, Yes], [14, 11, 32, 32, Yes], [14, 12, 46, 46, Yes], [14, 13, 34, 34, Yes], [14, 14, 36, 36, Yes], [14, 15, 38, 38, Yes], [14, 16, 40, 40, Yes], [14, 17, 0, 0, Yes], [14, 18, 2, 2, Yes], [14, 19, 4, 4, Yes], [14, 20, 6, 6, Yes], [14, 21, 8, 8, Yes], [14, 22, 10, 10, Yes], [14, 23, 12, 12, Yes], [15, 0, 44, 44, Yes], [15, 1, 46, 46, Yes], [15, 2, 16, 16, Yes], [15, 3, 18, 18, Yes], [15, 4, 20, 20, Yes], [15, 5, 22, 22, Yes], [15, 6, 24, 24, Yes], [15, 7, 26, 26, Yes], [15, 8, 28, 28, Yes], [15, 9, 30, 30, Yes], [15, 10, 32, 32, Yes], [15, 11, 34, 34, Yes], [15, 12, 0, 0, Yes], [15, 13, 36, 36, Yes], [15, 14, 38, 38, Yes], [15, 15, 40, 40, Yes], [15, 16, 42, 42, Yes], [15, 17, 2, 2, Yes], [15, 18, 4, 4, Yes], [15, 19, 6, 6, Yes], [15, 20, 8, 8, Yes], [15, 21, 10, 10, Yes], [15, 22, 12, 12, Yes], [15, 23, 14, 14, Yes], [16, 0, 46, 46, Yes], [16, 1, 0, 0, Yes], [16, 2, 18, 18, Yes], [16, 3, 20, 20, Yes], [16, 4, 22, 22, Yes], [16, 5, 24, 24, Yes], [16, 6, 26, 26, Yes], [16, 7, 28, 28, Yes], [16, 8, 30, 30, Yes], [16, 9, 32, 32, Yes], [16, 10, 34, 34, Yes], [16, 11, 36, 36, Yes], [16, 12, 2, 2, Yes], [16, 13, 38, 38, Yes], [16, 14, 40, 40, Yes], [16, 15, 42, 42, Yes], [16, 16, 44, 44, Yes], [16, 17, 4, 4, Yes], [16, 18, 6, 6, Yes], [16, 19, 8, 8, Yes], [16, 20, 10, 10, Yes], [16, 21, 12, 12, Yes], [16, 22, 14, 14, Yes], [16, 23, 16, 16, Yes], [17, 0, 6, 6, Yes], [17, 1, 8, 8, Yes], [17, 2, 26, 26, Yes], [17, 3, 28, 28, Yes], [17, 4, 30, 30, Yes], [17, 5, 32, 32, Yes], [17, 6, 34, 34, Yes], [17, 7, 36, 36, Yes], [17, 8, 38, 38, Yes], [17, 9, 40, 40, Yes], [17, 10, 42, 42, Yes], [17, 11, 44, 44, Yes], [17, 12, 10, 10, Yes], [17, 13, 46, 46, Yes], [17, 14, 0, 0, Yes], [17, 15, 2, 2, Yes], [17, 16, 4, 4, Yes], [17, 17, 12, 12, Yes], [17, 18, 14, 14, Yes], [17, 19, 16, 16, Yes], [17, 20, 18, 18, Yes], [17, 21, 20, 20, Yes], [17, 22, 22, 22, Yes], [17, 23, 24, 24, Yes], [18, 0, 8, 8, Yes], [18, 1, 10, 10, Yes], [18, 2, 28, 28, Yes], [18, 3, 30, 30, Yes], [18, 4, 32, 32, Yes], [18, 5, 34, 34, Yes], [18, 6, 36, 36, Yes], [18, 7, 38, 38, Yes], [18, 8, 40, 40, Yes], [18, 9, 42, 42, Yes], [18, 10, 44, 44, Yes], [18, 11, 46, 46, Yes], [18, 12, 12, 12, Yes], [18, 13, 0, 0, Yes], [18, 14, 2, 2, Yes], [18, 15, 4, 4, Yes], [18, 16, 6, 6, Yes], [18, 17, 14, 14, Yes], [18, 18, 16, 16, Yes], [18, 19, 18, 18, Yes], [18, 20, 20, 20, Yes], [18, 21, 22, 22, Yes], [18, 22, 24, 24, Yes], [18, 23, 26, 26, Yes], [19, 0, 10, 10, Yes], [19, 1, 12, 12, Yes], [19, 2, 30, 30, Yes], [19, 3, 32, 32, Yes], [19, 4, 34, 34, Yes], [19, 5, 36, 36, Yes], [19, 6, 38, 38, Yes], [19, 7, 40, 40, Yes], [19, 8, 42, 42, Yes], [19, 9, 44, 44, Yes], [19, 10, 46, 46, Yes], [19, 11, 0, 0, Yes], [19, 12, 14, 14, Yes], [19, 13, 2, 2, Yes], [19, 14, 4, 4, Yes], [19, 15, 6, 6, Yes], [19, 16, 8, 8, Yes], [19, 17, 16, 16, Yes], [19, 18, 18, 18, Yes], [19, 19, 20, 20, Yes], [19, 20, 22, 22, Yes], [19, 21, 24, 24, Yes], [19, 22, 26, 26, Yes], [19, 23, 28, 28, Yes], [20, 0, 12, 12, Yes], [20, 1, 14, 14, Yes], [20, 2, 32, 32, Yes], [20, 3, 34, 34, Yes], [20, 4, 36, 36, Yes], [20, 5, 38, 38, Yes], [20, 6, 40, 40, Yes], [20, 7, 42, 42, Yes], [20, 8, 44, 44, Yes], [20, 9, 46, 46, Yes], [20, 10, 0, 0, Yes], [20, 11, 2, 2, Yes], [20, 12, 16, 16, Yes], [20, 13, 4, 4, Yes], [20, 14, 6, 6, Yes], [20, 15, 8, 8, Yes], [20, 16, 10, 10, Yes], [20, 17, 18, 18, Yes], [20, 18, 20, 20, Yes], [20, 19, 22, 22, Yes], [20, 20, 24, 24, Yes], [20, 21, 26, 26, Yes], [20, 22, 28, 28, Yes], [20, 23, 30, 30, Yes], [21, 0, 14, 14, Yes], [21, 1, 16, 16, Yes], [21, 2, 34, 34, Yes], [21, 3, 36, 36, Yes], [21, 4, 38, 38, Yes], [21, 5, 40, 40, Yes], [21, 6, 42, 42, Yes], [21, 7, 44, 44, Yes], [21, 8, 46, 46, Yes], [21, 9, 0, 0, Yes], [21, 10, 2, 2, Yes], [21, 11, 4, 4, Yes], [21, 12, 18, 18, Yes], [21, 13, 6, 6, Yes], [21, 14, 8, 8, Yes], [21, 15, 10, 10, Yes], [21, 16, 12, 12, Yes], [21, 17, 20, 20, Yes], [21, 18, 22, 22, Yes], [21, 19, 24, 24, Yes], [21, 20, 26, 26, Yes], [21, 21, 28, 28, Yes], [21, 22, 30, 30, Yes], [21, 23, 32, 32, Yes], [22, 0, 16, 16, Yes], [22, 1, 18, 18, Yes], [22, 2, 36, 36, Yes], [22, 3, 38, 38, Yes], [22, 4, 40, 40, Yes], [22, 5, 42, 42, Yes], [22, 6, 44, 44, Yes], [22, 7, 46, 46, Yes], [22, 8, 0, 0, Yes], [22, 9, 2, 2, Yes], [22, 10, 4, 4, Yes], [22, 11, 6, 6, Yes], [22, 12, 20, 20, Yes], [22, 13, 8, 8, Yes], [22, 14, 10, 10, Yes], [22, 15, 12, 12, Yes], [22, 16, 14, 14, Yes], [22, 17, 22, 22, Yes], [22, 18, 24, 24, Yes], [22, 19, 26, 26, Yes], [22, 20, 28, 28, Yes], [22, 21, 30, 30, Yes], [22, 22, 32, 32, Yes], [22, 23, 34, 34, Yes], [23, 0, 18, 18, Yes], [23, 1, 20, 20, Yes], [23, 2, 38, 38, Yes], [23, 3, 40, 40, Yes], [23, 4, 42, 42, Yes], [23, 5, 44, 44, Yes], [23, 6, 46, 46, Yes], [23, 7, 0, 0, Yes], [23, 8, 2, 2, Yes], [23, 9, 4, 4, Yes], [23, 10, 6, 6, Yes], [23, 11, 8, 8, Yes], [23, 12, 22, 22, Yes], [23, 13, 10, 10, Yes], [23, 14, 12, 12, Yes], [23, 15, 14, 14, Yes], [23, 16, 16, 16, Yes], [23, 17, 24, 24, Yes], [23, 18, 26, 26, Yes], [23, 19, 28, 28, Yes], [23, 20, 30, 30, Yes], [23, 21, 32, 32, Yes], [23, 22, 34, 34, Yes], [23, 23, 36, 36, Yes]], Yes, Yes, Yes, Yes, No] Conditions: none The identity f(ab)=f(a)f(b) preserves products.If f(a)=f(b),a^-1b is in K and the cosets agree,giving injectivity.Every image value f(g) is hit by gK,giving surjectivity onto the IMAGE.Equal cardinalities do not replace these obligations or give surjectivity to the whole target. Step identifiers and dependenciesRule: Iso1_ALL_induced_product_pairs_injective_and_onto_IMAGE_separate · Uses: G001, S009 S011 · impliesBefore source roles = [[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46], 48, 24] After derived = [No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No] Conditions: none Only actual surjectivity identifies image with the whole codomain.The final No preserves general abstract/formal proof obligations,not denial of this complete finite isomorphism certificate or its pedagogical argument. Step identifiers and dependenciesRule: Iso1_image_VS_codomain_scope_and_finite_general_proof_boundary · Uses: G001, S010 ## Reverse checks - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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, 40, 42, 44, 46], 48, 24, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof) 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: 2dd7bfe440fdbfec86733bf0cfc65ef7e257366c56d0bf49a6c55e047df14a76 - candidate record SHA-256: bbe1ef189ed15fe923038461472916ef9093ac8f1b0155f4326023429df8dc62 - semantic review receipt SHA-256: 74c63a2c12285a3b2073f8d9f102bc2ba5fb7ed890c5b7c3b8afa94f032eda82 - question-first witness SHA-256: 6d56428610f8e86c45fd00a2ad3b27ad9c052956946f0b91d71a68536f79d95f - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0098.json)