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R0048-24968 · q-20261015-advance-20261114-r01-finite-group-homomorphism-kernel-0043
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, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).
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, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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]], source identity=0, target table=[[0, 1], [1, 0]], target identity=0, and total mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]. Row x/column y gives x*y in the respective table. Both groups qualify first, then every pair must satisfy f(x*y)=f(x)*f(y). A total map alone is not a homomorphism. Give (kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate).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.
Before
source roles = [kernel-normal-subgroup-audit]
After
derived = [[kernel, contains_identity, product_closed, inverse_closed, conjugate_rows, subgroup, normal, certificate]]
Conditions: none
List the full source field order without JavaScript,keeping original data,Boolean flags,mathematical objects and nonclaims distinct.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
source roles = [Kernel ORIGINAL SOURCE group, 64, [[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], [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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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]], 0]
After
derived = [38, [[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]], Yes]
Conditions: none
Validate the complete closed n-by-n table and actual identity label,not an assumed0.Mod44 allows32 and mod45 allows64;do not mix carriers or tables.
Rule: Group_complete_Cayley_signature_actual_identity · Uses: G001, S001
Before
source roles = [Kernel ORIGINAL SOURCE group, 38, FOR EVERY x,y,z in0..n-1]
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]], 54872, Yes]
Conditions: none
Entry(x,y) checks (xy)z=x(yz) for every z,covering all n^3 triples,not samples.The complete table is already displayed,so do not print millions of redundant lookups.
Rule: Group_ALL_associativity_triples_compact_complete_pair_matrix · Uses: G001, S002
Before
source roles = [Kernel ORIGINAL SOURCE group, 0]
After
derived = [[0, 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], [[0, 0, 0, 0], [1, 37, 0, 0], [2, 36, 0, 0], [3, 35, 0, 0], [4, 34, 0, 0], [5, 33, 0, 0], [6, 32, 0, 0], [7, 31, 0, 0], [8, 30, 0, 0], [9, 29, 0, 0], [10, 28, 0, 0], [11, 27, 0, 0], [12, 26, 0, 0], [13, 25, 0, 0], [14, 24, 0, 0], [15, 23, 0, 0], [16, 22, 0, 0], [17, 21, 0, 0], [18, 20, 0, 0], [19, 19, 0, 0], [20, 18, 0, 0], [21, 17, 0, 0], [22, 16, 0, 0], [23, 15, 0, 0], [24, 14, 0, 0], [25, 13, 0, 0], [26, 12, 0, 0], [27, 11, 0, 0], [28, 10, 0, 0], [29, 9, 0, 0], [30, 8, 0, 0], [31, 7, 0, 0], [32, 6, 0, 0], [33, 5, 0, 0], [34, 4, 0, 0], [35, 3, 0, 0], [36, 2, 0, 0], [37, 1, 0, 0]], Yes]
Conditions: none
Every left and right inverse must return the same actual identity;associativity gives uniqueness.This completes the group premises.
Rule: Group_ALL_two_sided_inverses · Uses: G001, S003
Before
source roles = [Kernel ORIGINAL TARGET group, 64, [[0, 1], [1, 0]], 0]
After
derived = [2, [[0, 0, 0, 0], [1, 1, 1, 1]], Yes]
Conditions: none
Validate the complete closed n-by-n table and actual identity label,not an assumed0.Mod44 allows32 and mod45 allows64;do not mix carriers or tables.
Rule: Group_complete_Cayley_signature_actual_identity · Uses: G001, S004
Before
source roles = [Kernel ORIGINAL TARGET group, 2, FOR EVERY x,y,z in0..n-1]
After
derived = [[[Yes, Yes], [Yes, Yes]], 8, Yes]
Conditions: none
Entry(x,y) checks (xy)z=x(yz) for every z,covering all n^3 triples,not samples.The complete table is already displayed,so do not print millions of redundant lookups.
Rule: Group_ALL_associativity_triples_compact_complete_pair_matrix · Uses: G001, S005
Before
source roles = [Kernel ORIGINAL TARGET group, 0]
After
derived = [[0, 1], [[0, 0, 0, 0], [1, 1, 0, 0]], Yes]
Conditions: none
Every left and right inverse must return the same actual identity;associativity gives uniqueness.This completes the group premises.
Rule: Group_ALL_two_sided_inverses · Uses: G001, S006
Before
source roles = [[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37], [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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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]], [[0, 1], [1, 0]], [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]]
After
derived = [[[0, 0, 0, 0, Yes], [0, 1, 1, 1, Yes], [0, 2, 0, 0, Yes], [0, 3, 1, 1, Yes], [0, 4, 0, 0, Yes], [0, 5, 1, 1, Yes], [0, 6, 0, 0, Yes], [0, 7, 1, 1, Yes], [0, 8, 0, 0, Yes], [0, 9, 1, 1, Yes], [0, 10, 0, 0, Yes], [0, 11, 1, 1, Yes], [0, 12, 0, 0, Yes], [0, 13, 1, 1, Yes], [0, 14, 0, 0, Yes], [0, 15, 1, 1, Yes], [0, 16, 0, 0, Yes], [0, 17, 1, 1, Yes], [0, 18, 0, 0, Yes], [0, 19, 1, 1, Yes], [0, 20, 0, 0, Yes], [0, 21, 1, 1, Yes], [0, 22, 0, 0, Yes], [0, 23, 1, 1, Yes], [0, 24, 0, 0, Yes], [0, 25, 1, 1, Yes], [0, 26, 0, 0, Yes], [0, 27, 1, 1, Yes], [0, 28, 0, 0, Yes], [0, 29, 1, 1, Yes], [0, 30, 0, 0, Yes], [0, 31, 1, 1, Yes], [0, 32, 0, 0, Yes], [0, 33, 1, 1, Yes], [0, 34, 0, 0, Yes], [0, 35, 1, 1, Yes], [0, 36, 0, 0, Yes], [0, 37, 1, 1, Yes], [1, 0, 1, 1, Yes], [1, 1, 0, 0, Yes], [1, 2, 1, 1, Yes], [1, 3, 0, 0, Yes], [1, 4, 1, 1, Yes], [1, 5, 0, 0, Yes], [1, 6, 1, 1, Yes], [1, 7, 0, 0, Yes], [1, 8, 1, 1, Yes], [1, 9, 0, 0, Yes], [1, 10, 1, 1, Yes], [1, 11, 0, 0, Yes], [1, 12, 1, 1, Yes], [1, 13, 0, 0, Yes], [1, 14, 1, 1, Yes], [1, 15, 0, 0, Yes], [1, 16, 1, 1, Yes], [1, 17, 0, 0, Yes], [1, 18, 1, 1, Yes], [1, 19, 0, 0, Yes], [1, 20, 1, 1, Yes], [1, 21, 0, 0, Yes], [1, 22, 1, 1, Yes], [1, 23, 0, 0, Yes], [1, 24, 1, 1, Yes], [1, 25, 0, 0, Yes], [1, 26, 1, 1, Yes], [1, 27, 0, 0, Yes], [1, 28, 1, 1, Yes], [1, 29, 0, 0, Yes], [1, 30, 1, 1, Yes], [1, 31, 0, 0, Yes], [1, 32, 1, 1, Yes], [1, 33, 0, 0, Yes], [1, 34, 1, 1, Yes], [1, 35, 0, 0, Yes], [1, 36, 1, 1, Yes], [1, 37, 0, 0, Yes], [2, 0, 0, 0, Yes], [2, 1, 1, 1, Yes], [2, 2, 0, 0, Yes], [2, 3, 1, 1, Yes], [2, 4, 0, 0, Yes], [2, 5, 1, 1, Yes], [2, 6, 0, 0, Yes], [2, 7, 1, 1, Yes], [2, 8, 0, 0, Yes], [2, 9, 1, 1, Yes], [2, 10, 0, 0, Yes], [2, 11, 1, 1, Yes], [2, 12, 0, 0, Yes], [2, 13, 1, 1, Yes], [2, 14, 0, 0, Yes], [2, 15, 1, 1, Yes], [2, 16, 0, 0, Yes], [2, 17, 1, 1, Yes], [2, 18, 0, 0, Yes], [2, 19, 1, 1, Yes], [2, 20, 0, 0, Yes], [2, 21, 1, 1, Yes], [2, 22, 0, 0, Yes], [2, 23, 1, 1, Yes], [2, 24, 0, 0, Yes], [2, 25, 1, 1, Yes], [2, 26, 0, 0, Yes], [2, 27, 1, 1, Yes], [2, 28, 0, 0, Yes], [2, 29, 1, 1, Yes], [2, 30, 0, 0, Yes], [2, 31, 1, 1, Yes], [2, 32, 0, 0, Yes], [2, 33, 1, 1, Yes], [2, 34, 0, 0, Yes], [2, 35, 1, 1, Yes], [2, 36, 0, 0, Yes], [2, 37, 1, 1, Yes], [3, 0, 1, 1, Yes], [3, 1, 0, 0, Yes], [3, 2, 1, 1, Yes], [3, 3, 0, 0, Yes], [3, 4, 1, 1, Yes], [3, 5, 0, 0, Yes], [3, 6, 1, 1, Yes], [3, 7, 0, 0, Yes], [3, 8, 1, 1, Yes], [3, 9, 0, 0, Yes], [3, 10, 1, 1, Yes], [3, 11, 0, 0, Yes], [3, 12, 1, 1, Yes], [3, 13, 0, 0, Yes], [3, 14, 1, 1, Yes], [3, 15, 0, 0, Yes], [3, 16, 1, 1, Yes], [3, 17, 0, 0, Yes], [3, 18, 1, 1, Yes], [3, 19, 0, 0, Yes], [3, 20, 1, 1, Yes], [3, 21, 0, 0, Yes], [3, 22, 1, 1, Yes], [3, 23, 0, 0, Yes], [3, 24, 1, 1, Yes], [3, 25, 0, 0, Yes], [3, 26, 1, 1, Yes], [3, 27, 0, 0, Yes], [3, 28, 1, 1, Yes], [3, 29, 0, 0, Yes], [3, 30, 1, 1, Yes], [3, 31, 0, 0, Yes], [3, 32, 1, 1, Yes], [3, 33, 0, 0, Yes], [3, 34, 1, 1, Yes], [3, 35, 0, 0, Yes], [3, 36, 1, 1, Yes], [3, 37, 0, 0, Yes], [4, 0, 0, 0, Yes], [4, 1, 1, 1, Yes], [4, 2, 0, 0, Yes], [4, 3, 1, 1, Yes], [4, 4, 0, 0, Yes], [4, 5, 1, 1, Yes], [4, 6, 0, 0, Yes], [4, 7, 1, 1, Yes], [4, 8, 0, 0, Yes], [4, 9, 1, 1, Yes], [4, 10, 0, 0, Yes], [4, 11, 1, 1, Yes], [4, 12, 0, 0, Yes], [4, 13, 1, 1, Yes], [4, 14, 0, 0, Yes], [4, 15, 1, 1, Yes], [4, 16, 0, 0, Yes], [4, 17, 1, 1, Yes], [4, 18, 0, 0, Yes], [4, 19, 1, 1, Yes], [4, 20, 0, 0, Yes], [4, 21, 1, 1, Yes], [4, 22, 0, 0, Yes], [4, 23, 1, 1, Yes], [4, 24, 0, 0, Yes], [4, 25, 1, 1, Yes], [4, 26, 0, 0, Yes], [4, 27, 1, 1, Yes], [4, 28, 0, 0, Yes], [4, 29, 1, 1, Yes], [4, 30, 0, 0, Yes], [4, 31, 1, 1, Yes], [4, 32, 0, 0, Yes], [4, 33, 1, 1, Yes], [4, 34, 0, 0, Yes], [4, 35, 1, 1, Yes], [4, 36, 0, 0, Yes], [4, 37, 1, 1, Yes], [5, 0, 1, 1, Yes], [5, 1, 0, 0, Yes], [5, 2, 1, 1, Yes], [5, 3, 0, 0, Yes], [5, 4, 1, 1, Yes], [5, 5, 0, 0, Yes], [5, 6, 1, 1, Yes], [5, 7, 0, 0, Yes], [5, 8, 1, 1, Yes], [5, 9, 0, 0, Yes], [5, 10, 1, 1, Yes], [5, 11, 0, 0, Yes], [5, 12, 1, 1, Yes], [5, 13, 0, 0, Yes], [5, 14, 1, 1, Yes], [5, 15, 0, 0, Yes], [5, 16, 1, 1, Yes], [5, 17, 0, 0, Yes], [5, 18, 1, 1, Yes], [5, 19, 0, 0, Yes], [5, 20, 1, 1, Yes], [5, 21, 0, 0, Yes], [5, 22, 1, 1, Yes], [5, 23, 0, 0, Yes], [5, 24, 1, 1, Yes], [5, 25, 0, 0, Yes], [5, 26, 1, 1, Yes], [5, 27, 0, 0, Yes], [5, 28, 1, 1, Yes], [5, 29, 0, 0, Yes], [5, 30, 1, 1, Yes], [5, 31, 0, 0, Yes], [5, 32, 1, 1, Yes], [5, 33, 0, 0, Yes], [5, 34, 1, 1, Yes], [5, 35, 0, 0, Yes], [5, 36, 1, 1, Yes], [5, 37, 0, 0, Yes], [6, 0, 0, 0, Yes], [6, 1, 1, 1, Yes], [6, 2, 0, 0, Yes], [6, 3, 1, 1, Yes], [6, 4, 0, 0, Yes], [6, 5, 1, 1, Yes], [6, 6, 0, 0, Yes], [6, 7, 1, 1, Yes], [6, 8, 0, 0, Yes], [6, 9, 1, 1, Yes], [6, 10, 0, 0, Yes], [6, 11, 1, 1, Yes], [6, 12, 0, 0, Yes], [6, 13, 1, 1, Yes], [6, 14, 0, 0, Yes], [6, 15, 1, 1, Yes], [6, 16, 0, 0, Yes], [6, 17, 1, 1, Yes], [6, 18, 0, 0, Yes], [6, 19, 1, 1, Yes], [6, 20, 0, 0, Yes], [6, 21, 1, 1, Yes], [6, 22, 0, 0, Yes], [6, 23, 1, 1, Yes], [6, 24, 0, 0, Yes], [6, 25, 1, 1, Yes], [6, 26, 0, 0, Yes], [6, 27, 1, 1, Yes], [6, 28, 0, 0, Yes], [6, 29, 1, 1, Yes], [6, 30, 0, 0, Yes], [6, 31, 1, 1, Yes], [6, 32, 0, 0, Yes], [6, 33, 1, 1, Yes], [6, 34, 0, 0, Yes], [6, 35, 1, 1, Yes], [6, 36, 0, 0, Yes], [6, 37, 1, 1, Yes], [7, 0, 1, 1, Yes], [7, 1, 0, 0, Yes], [7, 2, 1, 1, Yes], [7, 3, 0, 0, Yes], [7, 4, 1, 1, Yes], [7, 5, 0, 0, Yes], [7, 6, 1, 1, Yes], [7, 7, 0, 0, Yes], [7, 8, 1, 1, Yes], [7, 9, 0, 0, Yes], [7, 10, 1, 1, Yes], [7, 11, 0, 0, Yes], [7, 12, 1, 1, Yes], [7, 13, 0, 0, Yes], [7, 14, 1, 1, Yes], [7, 15, 0, 0, Yes], [7, 16, 1, 1, Yes], [7, 17, 0, 0, Yes], [7, 18, 1, 1, Yes], [7, 19, 0, 0, Yes], [7, 20, 1, 1, Yes], [7, 21, 0, 0, Yes], [7, 22, 1, 1, Yes], [7, 23, 0, 0, Yes], [7, 24, 1, 1, Yes], [7, 25, 0, 0, Yes], [7, 26, 1, 1, Yes], [7, 27, 0, 0, Yes], [7, 28, 1, 1, Yes], [7, 29, 0, 0, Yes], [7, 30, 1, 1, Yes], [7, 31, 0, 0, Yes], [7, 32, 1, 1, Yes], [7, 33, 0, 0, Yes], [7, 34, 1, 1, Yes], [7, 35, 0, 0, Yes], [7, 36, 1, 1, Yes], [7, 37, 0, 0, Yes], [8, 0, 0, 0, Yes], [8, 1, 1, 1, Yes], [8, 2, 0, 0, Yes], [8, 3, 1, 1, Yes], [8, 4, 0, 0, Yes], [8, 5, 1, 1, Yes], [8, 6, 0, 0, Yes], [8, 7, 1, 1, Yes], [8, 8, 0, 0, Yes], [8, 9, 1, 1, Yes], [8, 10, 0, 0, Yes], [8, 11, 1, 1, Yes], [8, 12, 0, 0, Yes], [8, 13, 1, 1, Yes], [8, 14, 0, 0, Yes], [8, 15, 1, 1, Yes], [8, 16, 0, 0, Yes], [8, 17, 1, 1, Yes], [8, 18, 0, 0, Yes], [8, 19, 1, 1, Yes], [8, 20, 0, 0, Yes], [8, 21, 1, 1, Yes], [8, 22, 0, 0, Yes], [8, 23, 1, 1, Yes], [8, 24, 0, 0, Yes], [8, 25, 1, 1, Yes], [8, 26, 0, 0, Yes], [8, 27, 1, 1, Yes], [8, 28, 0, 0, Yes], [8, 29, 1, 1, Yes], [8, 30, 0, 0, Yes], [8, 31, 1, 1, Yes], [8, 32, 0, 0, Yes], [8, 33, 1, 1, Yes], [8, 34, 0, 0, Yes], [8, 35, 1, 1, Yes], [8, 36, 0, 0, Yes], [8, 37, 1, 1, Yes], [9, 0, 1, 1, Yes], [9, 1, 0, 0, Yes], [9, 2, 1, 1, Yes], [9, 3, 0, 0, Yes], [9, 4, 1, 1, Yes], [9, 5, 0, 0, Yes], [9, 6, 1, 1, Yes], [9, 7, 0, 0, Yes], [9, 8, 1, 1, Yes], [9, 9, 0, 0, Yes], [9, 10, 1, 1, Yes], [9, 11, 0, 0, Yes], [9, 12, 1, 1, Yes], [9, 13, 0, 0, Yes], [9, 14, 1, 1, Yes], [9, 15, 0, 0, Yes], [9, 16, 1, 1, Yes], [9, 17, 0, 0, Yes], [9, 18, 1, 1, Yes], [9, 19, 0, 0, Yes], [9, 20, 1, 1, Yes], [9, 21, 0, 0, Yes], [9, 22, 1, 1, Yes], [9, 23, 0, 0, Yes], [9, 24, 1, 1, Yes], [9, 25, 0, 0, Yes], [9, 26, 1, 1, Yes], [9, 27, 0, 0, Yes], [9, 28, 1, 1, Yes], [9, 29, 0, 0, Yes], [9, 30, 1, 1, Yes], [9, 31, 0, 0, Yes], [9, 32, 1, 1, Yes], [9, 33, 0, 0, Yes], [9, 34, 1, 1, Yes], [9, 35, 0, 0, Yes], [9, 36, 1, 1, Yes], [9, 37, 0, 0, Yes], [10, 0, 0, 0, Yes], [10, 1, 1, 1, Yes], [10, 2, 0, 0, Yes], [10, 3, 1, 1, Yes], [10, 4, 0, 0, Yes], [10, 5, 1, 1, Yes], [10, 6, 0, 0, Yes], [10, 7, 1, 1, Yes], [10, 8, 0, 0, Yes], [10, 9, 1, 1, Yes], [10, 10, 0, 0, Yes], [10, 11, 1, 1, Yes], [10, 12, 0, 0, Yes], [10, 13, 1, 1, Yes], [10, 14, 0, 0, Yes], [10, 15, 1, 1, Yes], [10, 16, 0, 0, Yes], [10, 17, 1, 1, Yes], [10, 18, 0, 0, Yes], [10, 19, 1, 1, Yes], [10, 20, 0, 0, Yes], [10, 21, 1, 1, Yes], [10, 22, 0, 0, Yes], [10, 23, 1, 1, Yes], [10, 24, 0, 0, Yes], [10, 25, 1, 1, Yes], [10, 26, 0, 0, Yes], [10, 27, 1, 1, Yes], [10, 28, 0, 0, Yes], [10, 29, 1, 1, Yes], [10, 30, 0, 0, Yes], [10, 31, 1, 1, Yes], [10, 32, 0, 0, Yes], [10, 33, 1, 1, Yes], [10, 34, 0, 0, Yes], [10, 35, 1, 1, Yes], [10, 36, 0, 0, Yes], [10, 37, 1, 1, Yes], [11, 0, 1, 1, Yes], [11, 1, 0, 0, Yes], [11, 2, 1, 1, Yes], [11, 3, 0, 0, Yes], [11, 4, 1, 1, Yes], [11, 5, 0, 0, Yes], [11, 6, 1, 1, Yes], [11, 7, 0, 0, Yes], [11, 8, 1, 1, Yes], [11, 9, 0, 0, Yes], [11, 10, 1, 1, Yes], [11, 11, 0, 0, Yes], [11, 12, 1, 1, Yes], [11, 13, 0, 0, Yes], [11, 14, 1, 1, Yes], [11, 15, 0, 0, Yes], [11, 16, 1, 1, Yes], [11, 17, 0, 0, Yes], [11, 18, 1, 1, Yes], [11, 19, 0, 0, Yes], [11, 20, 1, 1, Yes], [11, 21, 0, 0, Yes], [11, 22, 1, 1, Yes], [11, 23, 0, 0, Yes], [11, 24, 1, 1, Yes], [11, 25, 0, 0, Yes], [11, 26, 1, 1, Yes], [11, 27, 0, 0, Yes], [11, 28, 1, 1, Yes], [11, 29, 0, 0, Yes], [11, 30, 1, 1, Yes], [11, 31, 0, 0, Yes], [11, 32, 1, 1, Yes], [11, 33, 0, 0, Yes], [11, 34, 1, 1, Yes], [11, 35, 0, 0, Yes], [11, 36, 1, 1, Yes], [11, 37, 0, 0, Yes], [12, 0, 0, 0, Yes], [12, 1, 1, 1, Yes], [12, 2, 0, 0, Yes], [12, 3, 1, 1, Yes], [12, 4, 0, 0, Yes], [12, 5, 1, 1, Yes], [12, 6, 0, 0, Yes], [12, 7, 1, 1, Yes], [12, 8, 0, 0, Yes], [12, 9, 1, 1, Yes], [12, 10, 0, 0, Yes], [12, 11, 1, 1, Yes], [12, 12, 0, 0, Yes], [12, 13, 1, 1, Yes], [12, 14, 0, 0, Yes], [12, 15, 1, 1, Yes], [12, 16, 0, 0, Yes], [12, 17, 1, 1, Yes], [12, 18, 0, 0, Yes], [12, 19, 1, 1, Yes], [12, 20, 0, 0, Yes], [12, 21, 1, 1, Yes], [12, 22, 0, 0, Yes], [12, 23, 1, 1, Yes], [12, 24, 0, 0, Yes], [12, 25, 1, 1, Yes], [12, 26, 0, 0, Yes], [12, 27, 1, 1, Yes], [12, 28, 0, 0, Yes], [12, 29, 1, 1, Yes], [12, 30, 0, 0, Yes], [12, 31, 1, 1, Yes], [12, 32, 0, 0, Yes], [12, 33, 1, 1, Yes], [12, 34, 0, 0, Yes], [12, 35, 1, 1, Yes], [12, 36, 0, 0, Yes], [12, 37, 1, 1, Yes], [13, 0, 1, 1, Yes], [13, 1, 0, 0, Yes], [13, 2, 1, 1, Yes], [13, 3, 0, 0, Yes], [13, 4, 1, 1, Yes], [13, 5, 0, 0, Yes], [13, 6, 1, 1, Yes], [13, 7, 0, 0, Yes], [13, 8, 1, 1, Yes], [13, 9, 0, 0, Yes], [13, 10, 1, 1, Yes], [13, 11, 0, 0, Yes], [13, 12, 1, 1, Yes], [13, 13, 0, 0, Yes], [13, 14, 1, 1, Yes], [13, 15, 0, 0, Yes], [13, 16, 1, 1, Yes], [13, 17, 0, 0, Yes], [13, 18, 1, 1, Yes], [13, 19, 0, 0, Yes], [13, 20, 1, 1, Yes], [13, 21, 0, 0, Yes], [13, 22, 1, 1, Yes], [13, 23, 0, 0, Yes], [13, 24, 1, 1, Yes], [13, 25, 0, 0, Yes], [13, 26, 1, 1, Yes], [13, 27, 0, 0, Yes], [13, 28, 1, 1, Yes], [13, 29, 0, 0, Yes], [13, 30, 1, 1, Yes], [13, 31, 0, 0, Yes], [13, 32, 1, 1, Yes], [13, 33, 0, 0, Yes], [13, 34, 1, 1, Yes], [13, 35, 0, 0, Yes], [13, 36, 1, 1, Yes], [13, 37, 0, 0, Yes], [14, 0, 0, 0, Yes], [14, 1, 1, 1, Yes], [14, 2, 0, 0, Yes], [14, 3, 1, 1, Yes], [14, 4, 0, 0, Yes], [14, 5, 1, 1, Yes], [14, 6, 0, 0, Yes], [14, 7, 1, 1, Yes], [14, 8, 0, 0, Yes], [14, 9, 1, 1, Yes], [14, 10, 0, 0, Yes], [14, 11, 1, 1, Yes], [14, 12, 0, 0, Yes], [14, 13, 1, 1, Yes], [14, 14, 0, 0, Yes], [14, 15, 1, 1, Yes], [14, 16, 0, 0, Yes], [14, 17, 1, 1, Yes], [14, 18, 0, 0, Yes], [14, 19, 1, 1, Yes], [14, 20, 0, 0, Yes], [14, 21, 1, 1, Yes], [14, 22, 0, 0, Yes], [14, 23, 1, 1, Yes], [14, 24, 0, 0, Yes], [14, 25, 1, 1, Yes], [14, 26, 0, 0, Yes], [14, 27, 1, 1, Yes], [14, 28, 0, 0, Yes], [14, 29, 1, 1, Yes], [14, 30, 0, 0, Yes], [14, 31, 1, 1, Yes], [14, 32, 0, 0, Yes], [14, 33, 1, 1, Yes], [14, 34, 0, 0, Yes], [14, 35, 1, 1, Yes], [14, 36, 0, 0, Yes], [14, 37, 1, 1, Yes], [15, 0, 1, 1, Yes], [15, 1, 0, 0, Yes], [15, 2, 1, 1, Yes], [15, 3, 0, 0, Yes], [15, 4, 1, 1, Yes], [15, 5, 0, 0, Yes], [15, 6, 1, 1, Yes], [15, 7, 0, 0, Yes], [15, 8, 1, 1, Yes], [15, 9, 0, 0, Yes], [15, 10, 1, 1, Yes], [15, 11, 0, 0, Yes], [15, 12, 1, 1, Yes], [15, 13, 0, 0, Yes], [15, 14, 1, 1, Yes], [15, 15, 0, 0, Yes], [15, 16, 1, 1, Yes], [15, 17, 0, 0, Yes], [15, 18, 1, 1, Yes], [15, 19, 0, 0, Yes], [15, 20, 1, 1, Yes], [15, 21, 0, 0, Yes], [15, 22, 1, 1, Yes], [15, 23, 0, 0, Yes], [15, 24, 1, 1, Yes], [15, 25, 0, 0, Yes], [15, 26, 1, 1, Yes], [15, 27, 0, 0, Yes], [15, 28, 1, 1, Yes], [15, 29, 0, 0, Yes], [15, 30, 1, 1, Yes], [15, 31, 0, 0, Yes], [15, 32, 1, 1, Yes], [15, 33, 0, 0, Yes], [15, 34, 1, 1, Yes], [15, 35, 0, 0, Yes], [15, 36, 1, 1, Yes], [15, 37, 0, 0, Yes], [16, 0, 0, 0, Yes], [16, 1, 1, 1, Yes], [16, 2, 0, 0, Yes], [16, 3, 1, 1, Yes], [16, 4, 0, 0, Yes], [16, 5, 1, 1, Yes], [16, 6, 0, 0, Yes], [16, 7, 1, 1, Yes], [16, 8, 0, 0, Yes], [16, 9, 1, 1, Yes], [16, 10, 0, 0, Yes], [16, 11, 1, 1, Yes], [16, 12, 0, 0, Yes], [16, 13, 1, 1, Yes], [16, 14, 0, 0, Yes], [16, 15, 1, 1, Yes], [16, 16, 0, 0, Yes], [16, 17, 1, 1, Yes], [16, 18, 0, 0, Yes], [16, 19, 1, 1, Yes], [16, 20, 0, 0, Yes], [16, 21, 1, 1, Yes], [16, 22, 0, 0, Yes], [16, 23, 1, 1, Yes], [16, 24, 0, 0, Yes], [16, 25, 1, 1, Yes], [16, 26, 0, 0, Yes], [16, 27, 1, 1, Yes], [16, 28, 0, 0, Yes], [16, 29, 1, 1, Yes], [16, 30, 0, 0, Yes], [16, 31, 1, 1, Yes], [16, 32, 0, 0, Yes], [16, 33, 1, 1, Yes], [16, 34, 0, 0, Yes], [16, 35, 1, 1, Yes], [16, 36, 0, 0, Yes], [16, 37, 1, 1, Yes], [17, 0, 1, 1, Yes], [17, 1, 0, 0, Yes], [17, 2, 1, 1, Yes], [17, 3, 0, 0, Yes], [17, 4, 1, 1, Yes], [17, 5, 0, 0, Yes], [17, 6, 1, 1, Yes], [17, 7, 0, 0, Yes], [17, 8, 1, 1, Yes], [17, 9, 0, 0, Yes], [17, 10, 1, 1, Yes], [17, 11, 0, 0, Yes], [17, 12, 1, 1, Yes], [17, 13, 0, 0, Yes], [17, 14, 1, 1, Yes], [17, 15, 0, 0, Yes], [17, 16, 1, 1, Yes], [17, 17, 0, 0, Yes], [17, 18, 1, 1, Yes], [17, 19, 0, 0, Yes], [17, 20, 1, 1, Yes], [17, 21, 0, 0, Yes], [17, 22, 1, 1, Yes], [17, 23, 0, 0, Yes], [17, 24, 1, 1, Yes], [17, 25, 0, 0, Yes], [17, 26, 1, 1, Yes], [17, 27, 0, 0, Yes], [17, 28, 1, 1, Yes], [17, 29, 0, 0, Yes], [17, 30, 1, 1, Yes], [17, 31, 0, 0, Yes], [17, 32, 1, 1, Yes], [17, 33, 0, 0, Yes], [17, 34, 1, 1, Yes], [17, 35, 0, 0, Yes], [17, 36, 1, 1, Yes], [17, 37, 0, 0, Yes], [18, 0, 0, 0, Yes], [18, 1, 1, 1, Yes], [18, 2, 0, 0, Yes], [18, 3, 1, 1, Yes], [18, 4, 0, 0, Yes], [18, 5, 1, 1, Yes], [18, 6, 0, 0, Yes], [18, 7, 1, 1, Yes], [18, 8, 0, 0, Yes], [18, 9, 1, 1, Yes], [18, 10, 0, 0, Yes], [18, 11, 1, 1, Yes], [18, 12, 0, 0, Yes], [18, 13, 1, 1, Yes], [18, 14, 0, 0, Yes], [18, 15, 1, 1, Yes], [18, 16, 0, 0, Yes], [18, 17, 1, 1, Yes], [18, 18, 0, 0, Yes], [18, 19, 1, 1, Yes], [18, 20, 0, 0, Yes], [18, 21, 1, 1, Yes], [18, 22, 0, 0, Yes], [18, 23, 1, 1, Yes], [18, 24, 0, 0, Yes], [18, 25, 1, 1, Yes], [18, 26, 0, 0, Yes], [18, 27, 1, 1, Yes], [18, 28, 0, 0, Yes], [18, 29, 1, 1, Yes], [18, 30, 0, 0, Yes], [18, 31, 1, 1, Yes], [18, 32, 0, 0, Yes], [18, 33, 1, 1, Yes], [18, 34, 0, 0, Yes], [18, 35, 1, 1, Yes], [18, 36, 0, 0, Yes], [18, 37, 1, 1, Yes], [19, 0, 1, 1, Yes], [19, 1, 0, 0, Yes], [19, 2, 1, 1, Yes], [19, 3, 0, 0, Yes], [19, 4, 1, 1, Yes], [19, 5, 0, 0, Yes], [19, 6, 1, 1, Yes], [19, 7, 0, 0, Yes], [19, 8, 1, 1, Yes], [19, 9, 0, 0, Yes], [19, 10, 1, 1, Yes], [19, 11, 0, 0, Yes], [19, 12, 1, 1, Yes], [19, 13, 0, 0, Yes], [19, 14, 1, 1, Yes], [19, 15, 0, 0, Yes], [19, 16, 1, 1, Yes], [19, 17, 0, 0, Yes], [19, 18, 1, 1, Yes], [19, 19, 0, 0, Yes], [19, 20, 1, 1, Yes], [19, 21, 0, 0, Yes], [19, 22, 1, 1, Yes], [19, 23, 0, 0, Yes], [19, 24, 1, 1, Yes], [19, 25, 0, 0, Yes], [19, 26, 1, 1, Yes], [19, 27, 0, 0, Yes], [19, 28, 1, 1, Yes], [19, 29, 0, 0, Yes], [19, 30, 1, 1, Yes], [19, 31, 0, 0, Yes], [19, 32, 1, 1, Yes], [19, 33, 0, 0, Yes], [19, 34, 1, 1, Yes], [19, 35, 0, 0, Yes], [19, 36, 1, 1, Yes], [19, 37, 0, 0, Yes], [20, 0, 0, 0, Yes], [20, 1, 1, 1, Yes], [20, 2, 0, 0, Yes], [20, 3, 1, 1, Yes], [20, 4, 0, 0, Yes], [20, 5, 1, 1, Yes], [20, 6, 0, 0, Yes], [20, 7, 1, 1, Yes], [20, 8, 0, 0, Yes], [20, 9, 1, 1, Yes], [20, 10, 0, 0, Yes], [20, 11, 1, 1, Yes], [20, 12, 0, 0, Yes], [20, 13, 1, 1, Yes], [20, 14, 0, 0, Yes], [20, 15, 1, 1, Yes], [20, 16, 0, 0, Yes], [20, 17, 1, 1, Yes], [20, 18, 0, 0, Yes], [20, 19, 1, 1, Yes], [20, 20, 0, 0, Yes], [20, 21, 1, 1, Yes], [20, 22, 0, 0, Yes], [20, 23, 1, 1, Yes], [20, 24, 0, 0, Yes], [20, 25, 1, 1, Yes], [20, 26, 0, 0, Yes], [20, 27, 1, 1, Yes], [20, 28, 0, 0, Yes], [20, 29, 1, 1, Yes], [20, 30, 0, 0, Yes], [20, 31, 1, 1, Yes], [20, 32, 0, 0, Yes], [20, 33, 1, 1, Yes], [20, 34, 0, 0, Yes], [20, 35, 1, 1, Yes], [20, 36, 0, 0, Yes], [20, 37, 1, 1, Yes], [21, 0, 1, 1, Yes], [21, 1, 0, 0, Yes], [21, 2, 1, 1, Yes], [21, 3, 0, 0, Yes], [21, 4, 1, 1, Yes], [21, 5, 0, 0, Yes], [21, 6, 1, 1, Yes], [21, 7, 0, 0, Yes], [21, 8, 1, 1, Yes], [21, 9, 0, 0, Yes], [21, 10, 1, 1, Yes], [21, 11, 0, 0, Yes], [21, 12, 1, 1, Yes], [21, 13, 0, 0, Yes], [21, 14, 1, 1, Yes], [21, 15, 0, 0, Yes], [21, 16, 1, 1, Yes], [21, 17, 0, 0, Yes], [21, 18, 1, 1, Yes], [21, 19, 0, 0, Yes], [21, 20, 1, 1, Yes], [21, 21, 0, 0, Yes], [21, 22, 1, 1, Yes], [21, 23, 0, 0, Yes], [21, 24, 1, 1, Yes], [21, 25, 0, 0, Yes], [21, 26, 1, 1, Yes], [21, 27, 0, 0, Yes], [21, 28, 1, 1, Yes], [21, 29, 0, 0, Yes], [21, 30, 1, 1, Yes], [21, 31, 0, 0, Yes], [21, 32, 1, 1, Yes], [21, 33, 0, 0, Yes], [21, 34, 1, 1, Yes], [21, 35, 0, 0, Yes], [21, 36, 1, 1, Yes], [21, 37, 0, 0, Yes], [22, 0, 0, 0, Yes], [22, 1, 1, 1, Yes], [22, 2, 0, 0, Yes], [22, 3, 1, 1, Yes], [22, 4, 0, 0, Yes], [22, 5, 1, 1, Yes], [22, 6, 0, 0, Yes], [22, 7, 1, 1, Yes], [22, 8, 0, 0, Yes], [22, 9, 1, 1, Yes], [22, 10, 0, 0, Yes], [22, 11, 1, 1, Yes], [22, 12, 0, 0, Yes], [22, 13, 1, 1, Yes], [22, 14, 0, 0, Yes], [22, 15, 1, 1, Yes], [22, 16, 0, 0, Yes], [22, 17, 1, 1, Yes], [22, 18, 0, 0, Yes], [22, 19, 1, 1, Yes], [22, 20, 0, 0, Yes], [22, 21, 1, 1, Yes], [22, 22, 0, 0, Yes], [22, 23, 1, 1, Yes], [22, 24, 0, 0, Yes], [22, 25, 1, 1, Yes], [22, 26, 0, 0, Yes], [22, 27, 1, 1, Yes], [22, 28, 0, 0, Yes], [22, 29, 1, 1, Yes], [22, 30, 0, 0, Yes], [22, 31, 1, 1, Yes], [22, 32, 0, 0, Yes], [22, 33, 1, 1, Yes], [22, 34, 0, 0, Yes], [22, 35, 1, 1, Yes], [22, 36, 0, 0, Yes], [22, 37, 1, 1, Yes], [23, 0, 1, 1, Yes], [23, 1, 0, 0, Yes], [23, 2, 1, 1, Yes], [23, 3, 0, 0, Yes], [23, 4, 1, 1, Yes], [23, 5, 0, 0, Yes], [23, 6, 1, 1, Yes], [23, 7, 0, 0, Yes], [23, 8, 1, 1, Yes], [23, 9, 0, 0, Yes], [23, 10, 1, 1, Yes], [23, 11, 0, 0, Yes], [23, 12, 1, 1, Yes], [23, 13, 0, 0, Yes], [23, 14, 1, 1, Yes], [23, 15, 0, 0, Yes], [23, 16, 1, 1, Yes], [23, 17, 0, 0, Yes], [23, 18, 1, 1, Yes], [23, 19, 0, 0, Yes], [23, 20, 1, 1, Yes], [23, 21, 0, 0, Yes], [23, 22, 1, 1, Yes], [23, 23, 0, 0, Yes], [23, 24, 1, 1, Yes], [23, 25, 0, 0, Yes], [23, 26, 1, 1, Yes], [23, 27, 0, 0, Yes], [23, 28, 1, 1, Yes], [23, 29, 0, 0, Yes], [23, 30, 1, 1, Yes], [23, 31, 0, 0, Yes], [23, 32, 1, 1, Yes], [23, 33, 0, 0, Yes], [23, 34, 1, 1, Yes], [23, 35, 0, 0, Yes], [23, 36, 1, 1, Yes], [23, 37, 0, 0, Yes], [24, 0, 0, 0, Yes], [24, 1, 1, 1, Yes], [24, 2, 0, 0, Yes], [24, 3, 1, 1, Yes], [24, 4, 0, 0, Yes], [24, 5, 1, 1, Yes], [24, 6, 0, 0, Yes], [24, 7, 1, 1, Yes], [24, 8, 0, 0, Yes], [24, 9, 1, 1, Yes], [24, 10, 0, 0, Yes], [24, 11, 1, 1, Yes], [24, 12, 0, 0, Yes], [24, 13, 1, 1, Yes], [24, 14, 0, 0, Yes], [24, 15, 1, 1, Yes], [24, 16, 0, 0, Yes], [24, 17, 1, 1, Yes], [24, 18, 0, 0, Yes], [24, 19, 1, 1, Yes], [24, 20, 0, 0, Yes], [24, 21, 1, 1, Yes], [24, 22, 0, 0, Yes], [24, 23, 1, 1, Yes], [24, 24, 0, 0, Yes], [24, 25, 1, 1, Yes], [24, 26, 0, 0, Yes], [24, 27, 1, 1, Yes], [24, 28, 0, 0, Yes], [24, 29, 1, 1, Yes], [24, 30, 0, 0, Yes], [24, 31, 1, 1, Yes], [24, 32, 0, 0, Yes], [24, 33, 1, 1, Yes], [24, 34, 0, 0, Yes], [24, 35, 1, 1, Yes], [24, 36, 0, 0, Yes], [24, 37, 1, 1, Yes], [25, 0, 1, 1, Yes], [25, 1, 0, 0, Yes], [25, 2, 1, 1, Yes], [25, 3, 0, 0, Yes], [25, 4, 1, 1, Yes], [25, 5, 0, 0, Yes], [25, 6, 1, 1, Yes], [25, 7, 0, 0, Yes], [25, 8, 1, 1, Yes], [25, 9, 0, 0, Yes], [25, 10, 1, 1, Yes], [25, 11, 0, 0, Yes], [25, 12, 1, 1, Yes], [25, 13, 0, 0, Yes], [25, 14, 1, 1, Yes], [25, 15, 0, 0, Yes], [25, 16, 1, 1, Yes], [25, 17, 0, 0, Yes], [25, 18, 1, 1, Yes], [25, 19, 0, 0, Yes], [25, 20, 1, 1, Yes], [25, 21, 0, 0, Yes], [25, 22, 1, 1, Yes], [25, 23, 0, 0, Yes], [25, 24, 1, 1, Yes], [25, 25, 0, 0, Yes], [25, 26, 1, 1, Yes], [25, 27, 0, 0, Yes], [25, 28, 1, 1, Yes], [25, 29, 0, 0, Yes], [25, 30, 1, 1, Yes], [25, 31, 0, 0, Yes], [25, 32, 1, 1, Yes], [25, 33, 0, 0, Yes], [25, 34, 1, 1, Yes], [25, 35, 0, 0, Yes], [25, 36, 1, 1, Yes], [25, 37, 0, 0, Yes], [26, 0, 0, 0, Yes], [26, 1, 1, 1, Yes], [26, 2, 0, 0, Yes], [26, 3, 1, 1, Yes], [26, 4, 0, 0, Yes], [26, 5, 1, 1, Yes], [26, 6, 0, 0, Yes], [26, 7, 1, 1, Yes], [26, 8, 0, 0, Yes], [26, 9, 1, 1, Yes], [26, 10, 0, 0, Yes], [26, 11, 1, 1, Yes], [26, 12, 0, 0, Yes], [26, 13, 1, 1, Yes], [26, 14, 0, 0, Yes], [26, 15, 1, 1, Yes], [26, 16, 0, 0, Yes], [26, 17, 1, 1, Yes], [26, 18, 0, 0, Yes], [26, 19, 1, 1, Yes], [26, 20, 0, 0, Yes], [26, 21, 1, 1, Yes], [26, 22, 0, 0, Yes], [26, 23, 1, 1, Yes], [26, 24, 0, 0, Yes], [26, 25, 1, 1, Yes], [26, 26, 0, 0, Yes], [26, 27, 1, 1, Yes], [26, 28, 0, 0, Yes], [26, 29, 1, 1, Yes], [26, 30, 0, 0, Yes], [26, 31, 1, 1, Yes], [26, 32, 0, 0, Yes], [26, 33, 1, 1, Yes], [26, 34, 0, 0, Yes], [26, 35, 1, 1, Yes], [26, 36, 0, 0, Yes], [26, 37, 1, 1, Yes], [27, 0, 1, 1, Yes], [27, 1, 0, 0, Yes], [27, 2, 1, 1, Yes], [27, 3, 0, 0, Yes], [27, 4, 1, 1, Yes], [27, 5, 0, 0, Yes], [27, 6, 1, 1, Yes], [27, 7, 0, 0, Yes], [27, 8, 1, 1, Yes], [27, 9, 0, 0, Yes], [27, 10, 1, 1, Yes], [27, 11, 0, 0, Yes], [27, 12, 1, 1, Yes], [27, 13, 0, 0, Yes], [27, 14, 1, 1, Yes], [27, 15, 0, 0, Yes], [27, 16, 1, 1, Yes], [27, 17, 0, 0, Yes], [27, 18, 1, 1, Yes], [27, 19, 0, 0, Yes], [27, 20, 1, 1, Yes], [27, 21, 0, 0, Yes], [27, 22, 1, 1, Yes], [27, 23, 0, 0, Yes], [27, 24, 1, 1, Yes], [27, 25, 0, 0, Yes], [27, 26, 1, 1, Yes], [27, 27, 0, 0, Yes], [27, 28, 1, 1, Yes], [27, 29, 0, 0, Yes], [27, 30, 1, 1, Yes], [27, 31, 0, 0, Yes], [27, 32, 1, 1, Yes], [27, 33, 0, 0, Yes], [27, 34, 1, 1, Yes], [27, 35, 0, 0, Yes], [27, 36, 1, 1, Yes], [27, 37, 0, 0, Yes], [28, 0, 0, 0, Yes], [28, 1, 1, 1, Yes], [28, 2, 0, 0, Yes], [28, 3, 1, 1, Yes], [28, 4, 0, 0, Yes], [28, 5, 1, 1, Yes], [28, 6, 0, 0, Yes], [28, 7, 1, 1, Yes], [28, 8, 0, 0, Yes], [28, 9, 1, 1, Yes], [28, 10, 0, 0, Yes], [28, 11, 1, 1, Yes], [28, 12, 0, 0, Yes], [28, 13, 1, 1, Yes], [28, 14, 0, 0, Yes], [28, 15, 1, 1, Yes], [28, 16, 0, 0, Yes], [28, 17, 1, 1, Yes], [28, 18, 0, 0, Yes], [28, 19, 1, 1, Yes], [28, 20, 0, 0, Yes], [28, 21, 1, 1, Yes], [28, 22, 0, 0, Yes], [28, 23, 1, 1, Yes], [28, 24, 0, 0, Yes], [28, 25, 1, 1, Yes], [28, 26, 0, 0, Yes], [28, 27, 1, 1, Yes], [28, 28, 0, 0, Yes], [28, 29, 1, 1, Yes], [28, 30, 0, 0, Yes], [28, 31, 1, 1, Yes], [28, 32, 0, 0, Yes], [28, 33, 1, 1, Yes], [28, 34, 0, 0, Yes], [28, 35, 1, 1, Yes], [28, 36, 0, 0, Yes], [28, 37, 1, 1, Yes], [29, 0, 1, 1, Yes], [29, 1, 0, 0, Yes], [29, 2, 1, 1, Yes], [29, 3, 0, 0, Yes], [29, 4, 1, 1, Yes], [29, 5, 0, 0, Yes], [29, 6, 1, 1, Yes], [29, 7, 0, 0, Yes], [29, 8, 1, 1, Yes], [29, 9, 0, 0, Yes], [29, 10, 1, 1, Yes], [29, 11, 0, 0, Yes], [29, 12, 1, 1, Yes], [29, 13, 0, 0, Yes], [29, 14, 1, 1, Yes], [29, 15, 0, 0, Yes], [29, 16, 1, 1, Yes], [29, 17, 0, 0, Yes], [29, 18, 1, 1, Yes], [29, 19, 0, 0, Yes], [29, 20, 1, 1, Yes], [29, 21, 0, 0, Yes], [29, 22, 1, 1, Yes], [29, 23, 0, 0, Yes], [29, 24, 1, 1, Yes], [29, 25, 0, 0, Yes], [29, 26, 1, 1, Yes], [29, 27, 0, 0, Yes], [29, 28, 1, 1, Yes], [29, 29, 0, 0, Yes], [29, 30, 1, 1, Yes], [29, 31, 0, 0, Yes], [29, 32, 1, 1, Yes], [29, 33, 0, 0, Yes], [29, 34, 1, 1, Yes], [29, 35, 0, 0, Yes], [29, 36, 1, 1, Yes], [29, 37, 0, 0, Yes], [30, 0, 0, 0, Yes], [30, 1, 1, 1, Yes], [30, 2, 0, 0, Yes], [30, 3, 1, 1, Yes], [30, 4, 0, 0, Yes], [30, 5, 1, 1, Yes], [30, 6, 0, 0, Yes], [30, 7, 1, 1, Yes], [30, 8, 0, 0, Yes], [30, 9, 1, 1, Yes], [30, 10, 0, 0, Yes], [30, 11, 1, 1, Yes], [30, 12, 0, 0, Yes], [30, 13, 1, 1, Yes], [30, 14, 0, 0, Yes], [30, 15, 1, 1, Yes], [30, 16, 0, 0, Yes], [30, 17, 1, 1, Yes], [30, 18, 0, 0, Yes], [30, 19, 1, 1, Yes], [30, 20, 0, 0, Yes], [30, 21, 1, 1, Yes], [30, 22, 0, 0, Yes], [30, 23, 1, 1, Yes], [30, 24, 0, 0, Yes], [30, 25, 1, 1, Yes], [30, 26, 0, 0, Yes], [30, 27, 1, 1, Yes], [30, 28, 0, 0, Yes], [30, 29, 1, 1, Yes], [30, 30, 0, 0, Yes], [30, 31, 1, 1, Yes], [30, 32, 0, 0, Yes], [30, 33, 1, 1, Yes], [30, 34, 0, 0, Yes], [30, 35, 1, 1, Yes], [30, 36, 0, 0, Yes], [30, 37, 1, 1, Yes], [31, 0, 1, 1, Yes], [31, 1, 0, 0, Yes], [31, 2, 1, 1, Yes], [31, 3, 0, 0, Yes], [31, 4, 1, 1, Yes], [31, 5, 0, 0, Yes], [31, 6, 1, 1, Yes], [31, 7, 0, 0, Yes], [31, 8, 1, 1, Yes], [31, 9, 0, 0, Yes], [31, 10, 1, 1, Yes], [31, 11, 0, 0, Yes], [31, 12, 1, 1, Yes], [31, 13, 0, 0, Yes], [31, 14, 1, 1, Yes], [31, 15, 0, 0, Yes], [31, 16, 1, 1, Yes], [31, 17, 0, 0, Yes], [31, 18, 1, 1, Yes], [31, 19, 0, 0, Yes], [31, 20, 1, 1, Yes], [31, 21, 0, 0, Yes], [31, 22, 1, 1, Yes], [31, 23, 0, 0, Yes], [31, 24, 1, 1, Yes], [31, 25, 0, 0, Yes], [31, 26, 1, 1, Yes], [31, 27, 0, 0, Yes], [31, 28, 1, 1, Yes], [31, 29, 0, 0, Yes], [31, 30, 1, 1, Yes], [31, 31, 0, 0, Yes], [31, 32, 1, 1, Yes], [31, 33, 0, 0, Yes], [31, 34, 1, 1, Yes], [31, 35, 0, 0, Yes], [31, 36, 1, 1, Yes], [31, 37, 0, 0, Yes], [32, 0, 0, 0, Yes], [32, 1, 1, 1, Yes], [32, 2, 0, 0, Yes], [32, 3, 1, 1, Yes], [32, 4, 0, 0, Yes], [32, 5, 1, 1, Yes], [32, 6, 0, 0, Yes], [32, 7, 1, 1, Yes], [32, 8, 0, 0, Yes], [32, 9, 1, 1, Yes], [32, 10, 0, 0, Yes], [32, 11, 1, 1, Yes], [32, 12, 0, 0, Yes], [32, 13, 1, 1, Yes], [32, 14, 0, 0, Yes], [32, 15, 1, 1, Yes], [32, 16, 0, 0, Yes], [32, 17, 1, 1, Yes], [32, 18, 0, 0, Yes], [32, 19, 1, 1, Yes], [32, 20, 0, 0, Yes], [32, 21, 1, 1, Yes], [32, 22, 0, 0, Yes], [32, 23, 1, 1, Yes], [32, 24, 0, 0, Yes], [32, 25, 1, 1, Yes], [32, 26, 0, 0, Yes], [32, 27, 1, 1, Yes], [32, 28, 0, 0, Yes], [32, 29, 1, 1, Yes], [32, 30, 0, 0, Yes], [32, 31, 1, 1, Yes], [32, 32, 0, 0, Yes], [32, 33, 1, 1, Yes], [32, 34, 0, 0, Yes], [32, 35, 1, 1, Yes], [32, 36, 0, 0, Yes], [32, 37, 1, 1, Yes], [33, 0, 1, 1, Yes], [33, 1, 0, 0, Yes], [33, 2, 1, 1, Yes], [33, 3, 0, 0, Yes], [33, 4, 1, 1, Yes], [33, 5, 0, 0, Yes], [33, 6, 1, 1, Yes], [33, 7, 0, 0, Yes], [33, 8, 1, 1, Yes], [33, 9, 0, 0, Yes], [33, 10, 1, 1, Yes], [33, 11, 0, 0, Yes], [33, 12, 1, 1, Yes], [33, 13, 0, 0, Yes], [33, 14, 1, 1, Yes], [33, 15, 0, 0, Yes], [33, 16, 1, 1, Yes], [33, 17, 0, 0, Yes], [33, 18, 1, 1, Yes], [33, 19, 0, 0, Yes], [33, 20, 1, 1, Yes], [33, 21, 0, 0, Yes], [33, 22, 1, 1, Yes], [33, 23, 0, 0, Yes], [33, 24, 1, 1, Yes], [33, 25, 0, 0, Yes], [33, 26, 1, 1, Yes], [33, 27, 0, 0, Yes], [33, 28, 1, 1, Yes], [33, 29, 0, 0, Yes], [33, 30, 1, 1, Yes], [33, 31, 0, 0, Yes], [33, 32, 1, 1, Yes], [33, 33, 0, 0, Yes], [33, 34, 1, 1, Yes], [33, 35, 0, 0, Yes], [33, 36, 1, 1, Yes], [33, 37, 0, 0, Yes], [34, 0, 0, 0, Yes], [34, 1, 1, 1, Yes], [34, 2, 0, 0, Yes], [34, 3, 1, 1, Yes], [34, 4, 0, 0, Yes], [34, 5, 1, 1, Yes], [34, 6, 0, 0, Yes], [34, 7, 1, 1, Yes], [34, 8, 0, 0, Yes], [34, 9, 1, 1, Yes], [34, 10, 0, 0, Yes], [34, 11, 1, 1, Yes], [34, 12, 0, 0, Yes], [34, 13, 1, 1, Yes], [34, 14, 0, 0, Yes], [34, 15, 1, 1, Yes], [34, 16, 0, 0, Yes], [34, 17, 1, 1, Yes], [34, 18, 0, 0, Yes], [34, 19, 1, 1, Yes], [34, 20, 0, 0, Yes], [34, 21, 1, 1, Yes], [34, 22, 0, 0, Yes], [34, 23, 1, 1, Yes], [34, 24, 0, 0, Yes], [34, 25, 1, 1, Yes], [34, 26, 0, 0, Yes], [34, 27, 1, 1, Yes], [34, 28, 0, 0, Yes], [34, 29, 1, 1, Yes], [34, 30, 0, 0, Yes], [34, 31, 1, 1, Yes], [34, 32, 0, 0, Yes], [34, 33, 1, 1, Yes], [34, 34, 0, 0, Yes], [34, 35, 1, 1, Yes], [34, 36, 0, 0, Yes], [34, 37, 1, 1, Yes], [35, 0, 1, 1, Yes], [35, 1, 0, 0, Yes], [35, 2, 1, 1, Yes], [35, 3, 0, 0, Yes], [35, 4, 1, 1, Yes], [35, 5, 0, 0, Yes], [35, 6, 1, 1, Yes], [35, 7, 0, 0, Yes], [35, 8, 1, 1, Yes], [35, 9, 0, 0, Yes], [35, 10, 1, 1, Yes], [35, 11, 0, 0, Yes], [35, 12, 1, 1, Yes], [35, 13, 0, 0, Yes], [35, 14, 1, 1, Yes], [35, 15, 0, 0, Yes], [35, 16, 1, 1, Yes], [35, 17, 0, 0, Yes], [35, 18, 1, 1, Yes], [35, 19, 0, 0, Yes], [35, 20, 1, 1, Yes], [35, 21, 0, 0, Yes], [35, 22, 1, 1, Yes], [35, 23, 0, 0, Yes], [35, 24, 1, 1, Yes], [35, 25, 0, 0, Yes], [35, 26, 1, 1, Yes], [35, 27, 0, 0, Yes], [35, 28, 1, 1, Yes], [35, 29, 0, 0, Yes], [35, 30, 1, 1, Yes], [35, 31, 0, 0, Yes], [35, 32, 1, 1, Yes], [35, 33, 0, 0, Yes], [35, 34, 1, 1, Yes], [35, 35, 0, 0, Yes], [35, 36, 1, 1, Yes], [35, 37, 0, 0, Yes], [36, 0, 0, 0, Yes], [36, 1, 1, 1, Yes], [36, 2, 0, 0, Yes], [36, 3, 1, 1, Yes], [36, 4, 0, 0, Yes], [36, 5, 1, 1, Yes], [36, 6, 0, 0, Yes], [36, 7, 1, 1, Yes], [36, 8, 0, 0, Yes], [36, 9, 1, 1, Yes], [36, 10, 0, 0, Yes], [36, 11, 1, 1, Yes], [36, 12, 0, 0, Yes], [36, 13, 1, 1, Yes], [36, 14, 0, 0, Yes], [36, 15, 1, 1, Yes], [36, 16, 0, 0, Yes], [36, 17, 1, 1, Yes], [36, 18, 0, 0, Yes], [36, 19, 1, 1, Yes], [36, 20, 0, 0, Yes], [36, 21, 1, 1, Yes], [36, 22, 0, 0, Yes], [36, 23, 1, 1, Yes], [36, 24, 0, 0, Yes], [36, 25, 1, 1, Yes], [36, 26, 0, 0, Yes], [36, 27, 1, 1, Yes], [36, 28, 0, 0, Yes], [36, 29, 1, 1, Yes], [36, 30, 0, 0, Yes], [36, 31, 1, 1, Yes], [36, 32, 0, 0, Yes], [36, 33, 1, 1, Yes], [36, 34, 0, 0, Yes], [36, 35, 1, 1, Yes], [36, 36, 0, 0, Yes], [36, 37, 1, 1, Yes], [37, 0, 1, 1, Yes], [37, 1, 0, 0, Yes], [37, 2, 1, 1, Yes], [37, 3, 0, 0, Yes], [37, 4, 1, 1, Yes], [37, 5, 0, 0, Yes], [37, 6, 1, 1, Yes], [37, 7, 0, 0, Yes], [37, 8, 1, 1, Yes], [37, 9, 0, 0, Yes], [37, 10, 1, 1, Yes], [37, 11, 0, 0, Yes], [37, 12, 1, 1, Yes], [37, 13, 0, 0, Yes], [37, 14, 1, 1, Yes], [37, 15, 0, 0, Yes], [37, 16, 1, 1, Yes], [37, 17, 0, 0, Yes], [37, 18, 1, 1, Yes], [37, 19, 0, 0, Yes], [37, 20, 1, 1, Yes], [37, 21, 0, 0, Yes], [37, 22, 1, 1, Yes], [37, 23, 0, 0, Yes], [37, 24, 1, 1, Yes], [37, 25, 0, 0, Yes], [37, 26, 1, 1, Yes], [37, 27, 0, 0, Yes], [37, 28, 1, 1, Yes], [37, 29, 0, 0, Yes], [37, 30, 1, 1, Yes], [37, 31, 0, 0, Yes], [37, 32, 1, 1, Yes], [37, 33, 0, 0, Yes], [37, 34, 1, 1, Yes], [37, 35, 0, 0, Yes], [37, 36, 1, 1, Yes], [37, 37, 0, 0, Yes]], Yes, null]
Conditions: none
For these same source and target groups,check f(xy)=f(x)f(y) for every pair and retain the first failed row/null.A total map need not be a homomorphism.Failure blocks kernel theorems;an internal fallback is not the true identity preimage.
Rule: Kernel_complete_TOTAL_map_ALL_product_pairs_before_theorems · Uses: G001, S007
Before
source roles = [[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], 0, 0]
After
derived = [[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 0, 0]
Conditions: none
The kernel is the complete preimage of the actual target identity.Cancel f(e)=f(e)^2 in the target group to obtain f(e)=e_target,so the true kernel is nonempty;do not assume identity0.
Rule: Kernel_actual_TARGET_identity_complete_preimage · Uses: G001, S008
Before
source roles = [Kernel ACTUAL identity preimage, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 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], [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, 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, 0, Yes], [4, 36, 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, 0, Yes], [6, 34, 2, Yes], [6, 36, 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, 0, Yes], [8, 32, 2, Yes], [8, 34, 4, Yes], [8, 36, 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, 0, Yes], [10, 30, 2, Yes], [10, 32, 4, Yes], [10, 34, 6, Yes], [10, 36, 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, 0, Yes], [12, 28, 2, Yes], [12, 30, 4, Yes], [12, 32, 6, Yes], [12, 34, 8, Yes], [12, 36, 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, 0, Yes], [14, 26, 2, Yes], [14, 28, 4, Yes], [14, 30, 6, Yes], [14, 32, 8, Yes], [14, 34, 10, Yes], [14, 36, 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, 0, Yes], [16, 24, 2, Yes], [16, 26, 4, Yes], [16, 28, 6, Yes], [16, 30, 8, Yes], [16, 32, 10, Yes], [16, 34, 12, Yes], [16, 36, 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, 0, Yes], [18, 22, 2, Yes], [18, 24, 4, Yes], [18, 26, 6, Yes], [18, 28, 8, Yes], [18, 30, 10, Yes], [18, 32, 12, Yes], [18, 34, 14, Yes], [18, 36, 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, 0, Yes], [20, 20, 2, Yes], [20, 22, 4, Yes], [20, 24, 6, Yes], [20, 26, 8, Yes], [20, 28, 10, Yes], [20, 30, 12, Yes], [20, 32, 14, Yes], [20, 34, 16, Yes], [20, 36, 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, 0, Yes], [22, 18, 2, Yes], [22, 20, 4, Yes], [22, 22, 6, Yes], [22, 24, 8, Yes], [22, 26, 10, Yes], [22, 28, 12, Yes], [22, 30, 14, Yes], [22, 32, 16, Yes], [22, 34, 18, Yes], [22, 36, 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, 0, Yes], [24, 16, 2, Yes], [24, 18, 4, Yes], [24, 20, 6, Yes], [24, 22, 8, Yes], [24, 24, 10, Yes], [24, 26, 12, Yes], [24, 28, 14, Yes], [24, 30, 16, Yes], [24, 32, 18, Yes], [24, 34, 20, Yes], [24, 36, 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, 0, Yes], [26, 14, 2, Yes], [26, 16, 4, Yes], [26, 18, 6, Yes], [26, 20, 8, Yes], [26, 22, 10, Yes], [26, 24, 12, Yes], [26, 26, 14, Yes], [26, 28, 16, Yes], [26, 30, 18, Yes], [26, 32, 20, Yes], [26, 34, 22, Yes], [26, 36, 24, Yes], [28, 0, 28, Yes], [28, 2, 30, Yes], [28, 4, 32, Yes], [28, 6, 34, Yes], [28, 8, 36, Yes], [28, 10, 0, Yes], [28, 12, 2, Yes], [28, 14, 4, Yes], [28, 16, 6, Yes], [28, 18, 8, Yes], [28, 20, 10, Yes], [28, 22, 12, Yes], [28, 24, 14, Yes], [28, 26, 16, Yes], [28, 28, 18, Yes], [28, 30, 20, Yes], [28, 32, 22, Yes], [28, 34, 24, Yes], [28, 36, 26, Yes], [30, 0, 30, Yes], [30, 2, 32, Yes], [30, 4, 34, Yes], [30, 6, 36, Yes], [30, 8, 0, Yes], [30, 10, 2, Yes], [30, 12, 4, Yes], [30, 14, 6, Yes], [30, 16, 8, Yes], [30, 18, 10, Yes], [30, 20, 12, Yes], [30, 22, 14, Yes], [30, 24, 16, Yes], [30, 26, 18, Yes], [30, 28, 20, Yes], [30, 30, 22, Yes], [30, 32, 24, Yes], [30, 34, 26, Yes], [30, 36, 28, Yes], [32, 0, 32, Yes], [32, 2, 34, Yes], [32, 4, 36, Yes], [32, 6, 0, Yes], [32, 8, 2, Yes], [32, 10, 4, Yes], [32, 12, 6, Yes], [32, 14, 8, Yes], [32, 16, 10, Yes], [32, 18, 12, Yes], [32, 20, 14, Yes], [32, 22, 16, Yes], [32, 24, 18, Yes], [32, 26, 20, Yes], [32, 28, 22, Yes], [32, 30, 24, Yes], [32, 32, 26, Yes], [32, 34, 28, Yes], [32, 36, 30, Yes], [34, 0, 34, Yes], [34, 2, 36, Yes], [34, 4, 0, Yes], [34, 6, 2, Yes], [34, 8, 4, Yes], [34, 10, 6, Yes], [34, 12, 8, Yes], [34, 14, 10, Yes], [34, 16, 12, Yes], [34, 18, 14, Yes], [34, 20, 16, Yes], [34, 22, 18, Yes], [34, 24, 20, Yes], [34, 26, 22, Yes], [34, 28, 24, Yes], [34, 30, 26, Yes], [34, 32, 28, Yes], [34, 34, 30, Yes], [34, 36, 32, Yes], [36, 0, 36, Yes], [36, 2, 0, Yes], [36, 4, 2, Yes], [36, 6, 4, Yes], [36, 8, 6, Yes], [36, 10, 8, Yes], [36, 12, 10, Yes], [36, 14, 12, Yes], [36, 16, 14, Yes], [36, 18, 16, Yes], [36, 20, 18, Yes], [36, 22, 20, Yes], [36, 24, 22, Yes], [36, 26, 24, Yes], [36, 28, 26, Yes], [36, 30, 28, Yes], [36, 32, 30, Yes], [36, 34, 32, Yes], [36, 36, 34, Yes]], Yes, [[0, 0, Yes], [2, 36, Yes], [4, 34, Yes], [6, 32, Yes], [8, 30, Yes], [10, 28, Yes], [12, 26, Yes], [14, 24, Yes], [16, 22, Yes], [18, 20, Yes], [20, 18, Yes], [22, 16, Yes], [24, 14, Yes], [26, 12, Yes], [28, 10, Yes], [30, 8, Yes], [32, 6, Yes], [34, 4, Yes], [36, 2, Yes]], Yes, Yes]
Conditions: none
H is nonempty and unique.Check identity membership,all products and every inverse separately;their conjunction qualifies a subgroup.Size divisibility or finiteness alone is insufficient.
Rule: Subgroup_identity_ALL_products_ALL_inverses_separate · Uses: G001, S009
Before
source roles = [Kernel ACTUAL identity preimage, 0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 0]
After
derived = [[[0, 0, 0, 0], [2, 2, 0, 2], [4, 4, 0, 4], [6, 6, 0, 6], [8, 8, 0, 8], [10, 10, 0, 10], [12, 12, 0, 12], [14, 14, 0, 14], [16, 16, 0, 16], [18, 18, 0, 18], [20, 20, 0, 20], [22, 22, 0, 22], [24, 24, 0, 24], [26, 26, 0, 26], [28, 28, 0, 28], [30, 30, 0, 30], [32, 32, 0, 32], [34, 34, 0, 34], [36, 36, 0, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S010
Before
source roles = [Kernel ACTUAL identity preimage, 1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 37]
After
derived = [[[0, 1, 37, 0], [2, 3, 37, 2], [4, 5, 37, 4], [6, 7, 37, 6], [8, 9, 37, 8], [10, 11, 37, 10], [12, 13, 37, 12], [14, 15, 37, 14], [16, 17, 37, 16], [18, 19, 37, 18], [20, 21, 37, 20], [22, 23, 37, 22], [24, 25, 37, 24], [26, 27, 37, 26], [28, 29, 37, 28], [30, 31, 37, 30], [32, 33, 37, 32], [34, 35, 37, 34], [36, 37, 37, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S011
Before
source roles = [Kernel ACTUAL identity preimage, 2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 36]
After
derived = [[[0, 2, 36, 0], [2, 4, 36, 2], [4, 6, 36, 4], [6, 8, 36, 6], [8, 10, 36, 8], [10, 12, 36, 10], [12, 14, 36, 12], [14, 16, 36, 14], [16, 18, 36, 16], [18, 20, 36, 18], [20, 22, 36, 20], [22, 24, 36, 22], [24, 26, 36, 24], [26, 28, 36, 26], [28, 30, 36, 28], [30, 32, 36, 30], [32, 34, 36, 32], [34, 36, 36, 34], [36, 0, 36, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S012
Before
source roles = [Kernel ACTUAL identity preimage, 3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 35]
After
derived = [[[0, 3, 35, 0], [2, 5, 35, 2], [4, 7, 35, 4], [6, 9, 35, 6], [8, 11, 35, 8], [10, 13, 35, 10], [12, 15, 35, 12], [14, 17, 35, 14], [16, 19, 35, 16], [18, 21, 35, 18], [20, 23, 35, 20], [22, 25, 35, 22], [24, 27, 35, 24], [26, 29, 35, 26], [28, 31, 35, 28], [30, 33, 35, 30], [32, 35, 35, 32], [34, 37, 35, 34], [36, 1, 35, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S013
Before
source roles = [Kernel ACTUAL identity preimage, 4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 34]
After
derived = [[[0, 4, 34, 0], [2, 6, 34, 2], [4, 8, 34, 4], [6, 10, 34, 6], [8, 12, 34, 8], [10, 14, 34, 10], [12, 16, 34, 12], [14, 18, 34, 14], [16, 20, 34, 16], [18, 22, 34, 18], [20, 24, 34, 20], [22, 26, 34, 22], [24, 28, 34, 24], [26, 30, 34, 26], [28, 32, 34, 28], [30, 34, 34, 30], [32, 36, 34, 32], [34, 0, 34, 34], [36, 2, 34, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S014
Before
source roles = [Kernel ACTUAL identity preimage, 5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 33]
After
derived = [[[0, 5, 33, 0], [2, 7, 33, 2], [4, 9, 33, 4], [6, 11, 33, 6], [8, 13, 33, 8], [10, 15, 33, 10], [12, 17, 33, 12], [14, 19, 33, 14], [16, 21, 33, 16], [18, 23, 33, 18], [20, 25, 33, 20], [22, 27, 33, 22], [24, 29, 33, 24], [26, 31, 33, 26], [28, 33, 33, 28], [30, 35, 33, 30], [32, 37, 33, 32], [34, 1, 33, 34], [36, 3, 33, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S015
Before
source roles = [Kernel ACTUAL identity preimage, 6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 32]
After
derived = [[[0, 6, 32, 0], [2, 8, 32, 2], [4, 10, 32, 4], [6, 12, 32, 6], [8, 14, 32, 8], [10, 16, 32, 10], [12, 18, 32, 12], [14, 20, 32, 14], [16, 22, 32, 16], [18, 24, 32, 18], [20, 26, 32, 20], [22, 28, 32, 22], [24, 30, 32, 24], [26, 32, 32, 26], [28, 34, 32, 28], [30, 36, 32, 30], [32, 0, 32, 32], [34, 2, 32, 34], [36, 4, 32, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S016
Before
source roles = [Kernel ACTUAL identity preimage, 7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 31]
After
derived = [[[0, 7, 31, 0], [2, 9, 31, 2], [4, 11, 31, 4], [6, 13, 31, 6], [8, 15, 31, 8], [10, 17, 31, 10], [12, 19, 31, 12], [14, 21, 31, 14], [16, 23, 31, 16], [18, 25, 31, 18], [20, 27, 31, 20], [22, 29, 31, 22], [24, 31, 31, 24], [26, 33, 31, 26], [28, 35, 31, 28], [30, 37, 31, 30], [32, 1, 31, 32], [34, 3, 31, 34], [36, 5, 31, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S017
Before
source roles = [Kernel ACTUAL identity preimage, 8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 30]
After
derived = [[[0, 8, 30, 0], [2, 10, 30, 2], [4, 12, 30, 4], [6, 14, 30, 6], [8, 16, 30, 8], [10, 18, 30, 10], [12, 20, 30, 12], [14, 22, 30, 14], [16, 24, 30, 16], [18, 26, 30, 18], [20, 28, 30, 20], [22, 30, 30, 22], [24, 32, 30, 24], [26, 34, 30, 26], [28, 36, 30, 28], [30, 0, 30, 30], [32, 2, 30, 32], [34, 4, 30, 34], [36, 6, 30, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S018
Before
source roles = [Kernel ACTUAL identity preimage, 9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 29]
After
derived = [[[0, 9, 29, 0], [2, 11, 29, 2], [4, 13, 29, 4], [6, 15, 29, 6], [8, 17, 29, 8], [10, 19, 29, 10], [12, 21, 29, 12], [14, 23, 29, 14], [16, 25, 29, 16], [18, 27, 29, 18], [20, 29, 29, 20], [22, 31, 29, 22], [24, 33, 29, 24], [26, 35, 29, 26], [28, 37, 29, 28], [30, 1, 29, 30], [32, 3, 29, 32], [34, 5, 29, 34], [36, 7, 29, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S019
Before
source roles = [Kernel ACTUAL identity preimage, 10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 28]
After
derived = [[[0, 10, 28, 0], [2, 12, 28, 2], [4, 14, 28, 4], [6, 16, 28, 6], [8, 18, 28, 8], [10, 20, 28, 10], [12, 22, 28, 12], [14, 24, 28, 14], [16, 26, 28, 16], [18, 28, 28, 18], [20, 30, 28, 20], [22, 32, 28, 22], [24, 34, 28, 24], [26, 36, 28, 26], [28, 0, 28, 28], [30, 2, 28, 30], [32, 4, 28, 32], [34, 6, 28, 34], [36, 8, 28, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S020
Before
source roles = [Kernel ACTUAL identity preimage, 11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 27]
After
derived = [[[0, 11, 27, 0], [2, 13, 27, 2], [4, 15, 27, 4], [6, 17, 27, 6], [8, 19, 27, 8], [10, 21, 27, 10], [12, 23, 27, 12], [14, 25, 27, 14], [16, 27, 27, 16], [18, 29, 27, 18], [20, 31, 27, 20], [22, 33, 27, 22], [24, 35, 27, 24], [26, 37, 27, 26], [28, 1, 27, 28], [30, 3, 27, 30], [32, 5, 27, 32], [34, 7, 27, 34], [36, 9, 27, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S021
Before
source roles = [Kernel ACTUAL identity preimage, 12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 26]
After
derived = [[[0, 12, 26, 0], [2, 14, 26, 2], [4, 16, 26, 4], [6, 18, 26, 6], [8, 20, 26, 8], [10, 22, 26, 10], [12, 24, 26, 12], [14, 26, 26, 14], [16, 28, 26, 16], [18, 30, 26, 18], [20, 32, 26, 20], [22, 34, 26, 22], [24, 36, 26, 24], [26, 0, 26, 26], [28, 2, 26, 28], [30, 4, 26, 30], [32, 6, 26, 32], [34, 8, 26, 34], [36, 10, 26, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S022
Before
source roles = [Kernel ACTUAL identity preimage, 13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 25]
After
derived = [[[0, 13, 25, 0], [2, 15, 25, 2], [4, 17, 25, 4], [6, 19, 25, 6], [8, 21, 25, 8], [10, 23, 25, 10], [12, 25, 25, 12], [14, 27, 25, 14], [16, 29, 25, 16], [18, 31, 25, 18], [20, 33, 25, 20], [22, 35, 25, 22], [24, 37, 25, 24], [26, 1, 25, 26], [28, 3, 25, 28], [30, 5, 25, 30], [32, 7, 25, 32], [34, 9, 25, 34], [36, 11, 25, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S023
Before
source roles = [Kernel ACTUAL identity preimage, 14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 24]
After
derived = [[[0, 14, 24, 0], [2, 16, 24, 2], [4, 18, 24, 4], [6, 20, 24, 6], [8, 22, 24, 8], [10, 24, 24, 10], [12, 26, 24, 12], [14, 28, 24, 14], [16, 30, 24, 16], [18, 32, 24, 18], [20, 34, 24, 20], [22, 36, 24, 22], [24, 0, 24, 24], [26, 2, 24, 26], [28, 4, 24, 28], [30, 6, 24, 30], [32, 8, 24, 32], [34, 10, 24, 34], [36, 12, 24, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S024
Before
source roles = [Kernel ACTUAL identity preimage, 15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 23]
After
derived = [[[0, 15, 23, 0], [2, 17, 23, 2], [4, 19, 23, 4], [6, 21, 23, 6], [8, 23, 23, 8], [10, 25, 23, 10], [12, 27, 23, 12], [14, 29, 23, 14], [16, 31, 23, 16], [18, 33, 23, 18], [20, 35, 23, 20], [22, 37, 23, 22], [24, 1, 23, 24], [26, 3, 23, 26], [28, 5, 23, 28], [30, 7, 23, 30], [32, 9, 23, 32], [34, 11, 23, 34], [36, 13, 23, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S025
Before
source roles = [Kernel ACTUAL identity preimage, 16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 22]
After
derived = [[[0, 16, 22, 0], [2, 18, 22, 2], [4, 20, 22, 4], [6, 22, 22, 6], [8, 24, 22, 8], [10, 26, 22, 10], [12, 28, 22, 12], [14, 30, 22, 14], [16, 32, 22, 16], [18, 34, 22, 18], [20, 36, 22, 20], [22, 0, 22, 22], [24, 2, 22, 24], [26, 4, 22, 26], [28, 6, 22, 28], [30, 8, 22, 30], [32, 10, 22, 32], [34, 12, 22, 34], [36, 14, 22, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S026
Before
source roles = [Kernel ACTUAL identity preimage, 17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 21]
After
derived = [[[0, 17, 21, 0], [2, 19, 21, 2], [4, 21, 21, 4], [6, 23, 21, 6], [8, 25, 21, 8], [10, 27, 21, 10], [12, 29, 21, 12], [14, 31, 21, 14], [16, 33, 21, 16], [18, 35, 21, 18], [20, 37, 21, 20], [22, 1, 21, 22], [24, 3, 21, 24], [26, 5, 21, 26], [28, 7, 21, 28], [30, 9, 21, 30], [32, 11, 21, 32], [34, 13, 21, 34], [36, 15, 21, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S027
Before
source roles = [Kernel ACTUAL identity preimage, 18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 20]
After
derived = [[[0, 18, 20, 0], [2, 20, 20, 2], [4, 22, 20, 4], [6, 24, 20, 6], [8, 26, 20, 8], [10, 28, 20, 10], [12, 30, 20, 12], [14, 32, 20, 14], [16, 34, 20, 16], [18, 36, 20, 18], [20, 0, 20, 20], [22, 2, 20, 22], [24, 4, 20, 24], [26, 6, 20, 26], [28, 8, 20, 28], [30, 10, 20, 30], [32, 12, 20, 32], [34, 14, 20, 34], [36, 16, 20, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S028
Before
source roles = [Kernel ACTUAL identity preimage, 19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 19]
After
derived = [[[0, 19, 19, 0], [2, 21, 19, 2], [4, 23, 19, 4], [6, 25, 19, 6], [8, 27, 19, 8], [10, 29, 19, 10], [12, 31, 19, 12], [14, 33, 19, 14], [16, 35, 19, 16], [18, 37, 19, 18], [20, 1, 19, 20], [22, 3, 19, 22], [24, 5, 19, 24], [26, 7, 19, 26], [28, 9, 19, 28], [30, 11, 19, 30], [32, 13, 19, 32], [34, 15, 19, 34], [36, 17, 19, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S029
Before
source roles = [Kernel ACTUAL identity preimage, 20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 18]
After
derived = [[[0, 20, 18, 0], [2, 22, 18, 2], [4, 24, 18, 4], [6, 26, 18, 6], [8, 28, 18, 8], [10, 30, 18, 10], [12, 32, 18, 12], [14, 34, 18, 14], [16, 36, 18, 16], [18, 0, 18, 18], [20, 2, 18, 20], [22, 4, 18, 22], [24, 6, 18, 24], [26, 8, 18, 26], [28, 10, 18, 28], [30, 12, 18, 30], [32, 14, 18, 32], [34, 16, 18, 34], [36, 18, 18, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S030
Before
source roles = [Kernel ACTUAL identity preimage, 21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 17]
After
derived = [[[0, 21, 17, 0], [2, 23, 17, 2], [4, 25, 17, 4], [6, 27, 17, 6], [8, 29, 17, 8], [10, 31, 17, 10], [12, 33, 17, 12], [14, 35, 17, 14], [16, 37, 17, 16], [18, 1, 17, 18], [20, 3, 17, 20], [22, 5, 17, 22], [24, 7, 17, 24], [26, 9, 17, 26], [28, 11, 17, 28], [30, 13, 17, 30], [32, 15, 17, 32], [34, 17, 17, 34], [36, 19, 17, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S031
Before
source roles = [Kernel ACTUAL identity preimage, 22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 16]
After
derived = [[[0, 22, 16, 0], [2, 24, 16, 2], [4, 26, 16, 4], [6, 28, 16, 6], [8, 30, 16, 8], [10, 32, 16, 10], [12, 34, 16, 12], [14, 36, 16, 14], [16, 0, 16, 16], [18, 2, 16, 18], [20, 4, 16, 20], [22, 6, 16, 22], [24, 8, 16, 24], [26, 10, 16, 26], [28, 12, 16, 28], [30, 14, 16, 30], [32, 16, 16, 32], [34, 18, 16, 34], [36, 20, 16, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S032
Before
source roles = [Kernel ACTUAL identity preimage, 23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 15]
After
derived = [[[0, 23, 15, 0], [2, 25, 15, 2], [4, 27, 15, 4], [6, 29, 15, 6], [8, 31, 15, 8], [10, 33, 15, 10], [12, 35, 15, 12], [14, 37, 15, 14], [16, 1, 15, 16], [18, 3, 15, 18], [20, 5, 15, 20], [22, 7, 15, 22], [24, 9, 15, 24], [26, 11, 15, 26], [28, 13, 15, 28], [30, 15, 15, 30], [32, 17, 15, 32], [34, 19, 15, 34], [36, 21, 15, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S033
Before
source roles = [Kernel ACTUAL identity preimage, 24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 14]
After
derived = [[[0, 24, 14, 0], [2, 26, 14, 2], [4, 28, 14, 4], [6, 30, 14, 6], [8, 32, 14, 8], [10, 34, 14, 10], [12, 36, 14, 12], [14, 0, 14, 14], [16, 2, 14, 16], [18, 4, 14, 18], [20, 6, 14, 20], [22, 8, 14, 22], [24, 10, 14, 24], [26, 12, 14, 26], [28, 14, 14, 28], [30, 16, 14, 30], [32, 18, 14, 32], [34, 20, 14, 34], [36, 22, 14, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S034
Before
source roles = [Kernel ACTUAL identity preimage, 25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 13]
After
derived = [[[0, 25, 13, 0], [2, 27, 13, 2], [4, 29, 13, 4], [6, 31, 13, 6], [8, 33, 13, 8], [10, 35, 13, 10], [12, 37, 13, 12], [14, 1, 13, 14], [16, 3, 13, 16], [18, 5, 13, 18], [20, 7, 13, 20], [22, 9, 13, 22], [24, 11, 13, 24], [26, 13, 13, 26], [28, 15, 13, 28], [30, 17, 13, 30], [32, 19, 13, 32], [34, 21, 13, 34], [36, 23, 13, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S035
Before
source roles = [Kernel ACTUAL identity preimage, 26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 12]
After
derived = [[[0, 26, 12, 0], [2, 28, 12, 2], [4, 30, 12, 4], [6, 32, 12, 6], [8, 34, 12, 8], [10, 36, 12, 10], [12, 0, 12, 12], [14, 2, 12, 14], [16, 4, 12, 16], [18, 6, 12, 18], [20, 8, 12, 20], [22, 10, 12, 22], [24, 12, 12, 24], [26, 14, 12, 26], [28, 16, 12, 28], [30, 18, 12, 30], [32, 20, 12, 32], [34, 22, 12, 34], [36, 24, 12, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S036
Before
source roles = [Kernel ACTUAL identity preimage, 27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 11]
After
derived = [[[0, 27, 11, 0], [2, 29, 11, 2], [4, 31, 11, 4], [6, 33, 11, 6], [8, 35, 11, 8], [10, 37, 11, 10], [12, 1, 11, 12], [14, 3, 11, 14], [16, 5, 11, 16], [18, 7, 11, 18], [20, 9, 11, 20], [22, 11, 11, 22], [24, 13, 11, 24], [26, 15, 11, 26], [28, 17, 11, 28], [30, 19, 11, 30], [32, 21, 11, 32], [34, 23, 11, 34], [36, 25, 11, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S037
Before
source roles = [Kernel ACTUAL identity preimage, 28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 10]
After
derived = [[[0, 28, 10, 0], [2, 30, 10, 2], [4, 32, 10, 4], [6, 34, 10, 6], [8, 36, 10, 8], [10, 0, 10, 10], [12, 2, 10, 12], [14, 4, 10, 14], [16, 6, 10, 16], [18, 8, 10, 18], [20, 10, 10, 20], [22, 12, 10, 22], [24, 14, 10, 24], [26, 16, 10, 26], [28, 18, 10, 28], [30, 20, 10, 30], [32, 22, 10, 32], [34, 24, 10, 34], [36, 26, 10, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S038
Before
source roles = [Kernel ACTUAL identity preimage, 29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 9]
After
derived = [[[0, 29, 9, 0], [2, 31, 9, 2], [4, 33, 9, 4], [6, 35, 9, 6], [8, 37, 9, 8], [10, 1, 9, 10], [12, 3, 9, 12], [14, 5, 9, 14], [16, 7, 9, 16], [18, 9, 9, 18], [20, 11, 9, 20], [22, 13, 9, 22], [24, 15, 9, 24], [26, 17, 9, 26], [28, 19, 9, 28], [30, 21, 9, 30], [32, 23, 9, 32], [34, 25, 9, 34], [36, 27, 9, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S039
Before
source roles = [Kernel ACTUAL identity preimage, 30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 8]
After
derived = [[[0, 30, 8, 0], [2, 32, 8, 2], [4, 34, 8, 4], [6, 36, 8, 6], [8, 0, 8, 8], [10, 2, 8, 10], [12, 4, 8, 12], [14, 6, 8, 14], [16, 8, 8, 16], [18, 10, 8, 18], [20, 12, 8, 20], [22, 14, 8, 22], [24, 16, 8, 24], [26, 18, 8, 26], [28, 20, 8, 28], [30, 22, 8, 30], [32, 24, 8, 32], [34, 26, 8, 34], [36, 28, 8, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S040
Before
source roles = [Kernel ACTUAL identity preimage, 31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 7]
After
derived = [[[0, 31, 7, 0], [2, 33, 7, 2], [4, 35, 7, 4], [6, 37, 7, 6], [8, 1, 7, 8], [10, 3, 7, 10], [12, 5, 7, 12], [14, 7, 7, 14], [16, 9, 7, 16], [18, 11, 7, 18], [20, 13, 7, 20], [22, 15, 7, 22], [24, 17, 7, 24], [26, 19, 7, 26], [28, 21, 7, 28], [30, 23, 7, 30], [32, 25, 7, 32], [34, 27, 7, 34], [36, 29, 7, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S041
Before
source roles = [Kernel ACTUAL identity preimage, 32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 6]
After
derived = [[[0, 32, 6, 0], [2, 34, 6, 2], [4, 36, 6, 4], [6, 0, 6, 6], [8, 2, 6, 8], [10, 4, 6, 10], [12, 6, 6, 12], [14, 8, 6, 14], [16, 10, 6, 16], [18, 12, 6, 18], [20, 14, 6, 20], [22, 16, 6, 22], [24, 18, 6, 24], [26, 20, 6, 26], [28, 22, 6, 28], [30, 24, 6, 30], [32, 26, 6, 32], [34, 28, 6, 34], [36, 30, 6, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S042
Before
source roles = [Kernel ACTUAL identity preimage, 33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 5]
After
derived = [[[0, 33, 5, 0], [2, 35, 5, 2], [4, 37, 5, 4], [6, 1, 5, 6], [8, 3, 5, 8], [10, 5, 5, 10], [12, 7, 5, 12], [14, 9, 5, 14], [16, 11, 5, 16], [18, 13, 5, 18], [20, 15, 5, 20], [22, 17, 5, 22], [24, 19, 5, 24], [26, 21, 5, 26], [28, 23, 5, 28], [30, 25, 5, 30], [32, 27, 5, 32], [34, 29, 5, 34], [36, 31, 5, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S043
Before
source roles = [Kernel ACTUAL identity preimage, 34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 4]
After
derived = [[[0, 34, 4, 0], [2, 36, 4, 2], [4, 0, 4, 4], [6, 2, 4, 6], [8, 4, 4, 8], [10, 6, 4, 10], [12, 8, 4, 12], [14, 10, 4, 14], [16, 12, 4, 16], [18, 14, 4, 18], [20, 16, 4, 20], [22, 18, 4, 22], [24, 20, 4, 24], [26, 22, 4, 26], [28, 24, 4, 28], [30, 26, 4, 30], [32, 28, 4, 32], [34, 30, 4, 34], [36, 32, 4, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S044
Before
source roles = [Kernel ACTUAL identity preimage, 35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 3]
After
derived = [[[0, 35, 3, 0], [2, 37, 3, 2], [4, 1, 3, 4], [6, 3, 3, 6], [8, 5, 3, 8], [10, 7, 3, 10], [12, 9, 3, 12], [14, 11, 3, 14], [16, 13, 3, 16], [18, 15, 3, 18], [20, 17, 3, 20], [22, 19, 3, 22], [24, 21, 3, 24], [26, 23, 3, 26], [28, 25, 3, 28], [30, 27, 3, 30], [32, 29, 3, 32], [34, 31, 3, 34], [36, 33, 3, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S045
Before
source roles = [Kernel ACTUAL identity preimage, 36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 2]
After
derived = [[[0, 36, 2, 0], [2, 0, 2, 2], [4, 2, 2, 4], [6, 4, 2, 6], [8, 6, 2, 8], [10, 8, 2, 10], [12, 10, 2, 12], [14, 12, 2, 14], [16, 14, 2, 16], [18, 16, 2, 18], [20, 18, 2, 20], [22, 20, 2, 22], [24, 22, 2, 24], [26, 24, 2, 26], [28, 26, 2, 28], [30, 28, 2, 30], [32, 30, 2, 32], [34, 32, 2, 34], [36, 34, 2, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S046
Before
source roles = [Kernel ACTUAL identity preimage, 37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], 1]
After
derived = [[[0, 37, 1, 0], [2, 1, 1, 2], [4, 3, 1, 4], [6, 5, 1, 6], [8, 7, 1, 8], [10, 9, 1, 10], [12, 11, 1, 12], [14, 13, 1, 14], [16, 15, 1, 16], [18, 17, 1, 18], [20, 19, 1, 20], [22, 21, 1, 22], [24, 23, 1, 24], [26, 25, 1, 26], [28, 27, 1, 28], [30, 29, 1, 30], [32, 31, 1, 32], [34, 33, 1, 34], [36, 35, 1, 36]], [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]
Conditions: none
In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators.
Rule: Subgroup_EVERY_g_entire_conjugate_set · Uses: G001, S047
Before
source roles = [[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes, 0]
After
derived = [f(kl)=f(k)f(l)=e, f(k^-1)=f(k)^-1=e, f(gkg^-1)=f(g)e*f(g)^-1=e]
Conditions: none
Products,inverses and every conjugate map to target identity,giving the general closure/normality argument alongside this complete finite audit.No quotient/isomorphism is claimed.
Rule: Kernel_GENERAL_closure_normality_from_homomorphism · Uses: G001, S048
S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.([0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes, Yes, Yes, [[0, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [1, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [2, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [3, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [4, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [5, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [6, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [7, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [8, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [9, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [10, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [11, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [12, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [13, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [14, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [15, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [16, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [17, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [18, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [19, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [20, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [21, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [22, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [23, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [24, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [25, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [26, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [27, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [28, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [29, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [30, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [31, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [32, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [33, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [34, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [35, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [36, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes], [37, [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36], Yes]], Yes, Yes, identity_preimage_is_a_normal_subgroup_by_complete_audit)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 39d7326e071b4630b791548eac09b3b539ac165d7cd572e7ea995974a5ba6332candidate record SHA-256: 5fbbd6e52057623de10b749df0b813f0929bec6eb882b924d97f4c3c6ebce090semantic review receipt SHA-256: 049a9e07e69fead88d93b34f0937b9470a06d71c1a8d097058fc83554529bdd0question-first witness SHA-256: 0c71ebb418f3de347d232d159fc25e93036f4ab9166bcfb5ee5b4be86d5c55b9Lean: NOT_RUN · human review: NOT_PERFORMED