from sympy.combinatorics.fp_groups import FpGroup from sympy.combinatorics.coset_table import (CosetTable, coset_enumeration_r, coset_enumeration_c) from sympy.combinatorics.coset_table import modified_coset_enumeration_r from sympy.combinatorics.free_groups import free_group from sympy.testing.pytest import slow """ References ========== [1] Holt, D., Eick, B., O'Brien, E. "Handbook of Computational Group Theory" [2] John J. Cannon; Lucien A. Dimino; George Havas; Jane M. Watson Mathematics of Computation, Vol. 27, No. 123. (Jul., 1973), pp. 463-490. "Implementation and Analysis of the Todd-Coxeter Algorithm" """ def test_scan_1(): # Example 5.1 from [1] F, x, y = free_group("x, y") f = FpGroup(F, [x**3, y**3, x**-1*y**-1*x*y]) c = CosetTable(f, [x]) c.scan_and_fill(0, x) assert c.table == [[0, 0, None, None]] assert c.p == [0] assert c.n == 1 assert c.omega == [0] c.scan_and_fill(0, x**3) assert c.table == [[0, 0, None, None]] assert c.p == [0] assert c.n == 1 assert c.omega == [0] c.scan_and_fill(0, y**3) assert c.table == [[0, 0, 1, 2], [None, None, 2, 0], [None, None, 0, 1]] assert c.p == [0, 1, 2] assert c.n == 3 assert c.omega == [0, 1, 2] c.scan_and_fill(0, x**-1*y**-1*x*y) assert c.table == [[0, 0, 1, 2], [None, None, 2, 0], [2, 2, 0, 1]] assert c.p == [0, 1, 2] assert c.n == 3 assert c.omega == [0, 1, 2] c.scan_and_fill(1, x**3) assert c.table == [[0, 0, 1, 2], [3, 4, 2, 0], [2, 2, 0, 1], \ [4, 1, None, None], [1, 3, None, None]] assert c.p == [0, 1, 2, 3, 4] assert c.n == 5 assert c.omega == [0, 1, 2, 3, 4] c.scan_and_fill(1, y**3) assert c.table == [[0, 0, 1, 2], [3, 4, 2, 0], [2, 2, 0, 1], \ [4, 1, None, None], [1, 3, None, None]] assert c.p == [0, 1, 2, 3, 4] assert c.n == 5 assert c.omega == [0, 1, 2, 3, 4] c.scan_and_fill(1, x**-1*y**-1*x*y) assert c.table == [[0, 0, 1, 2], [1, 1, 2, 0], [2, 2, 0, 1], \ [None, 1, None, None], [1, 3, None, None]] assert c.p == [0, 1, 2, 1, 1] assert c.n == 3 assert c.omega == [0, 1, 2] # Example 5.2 from [1] f = FpGroup(F, [x**2, y**3, (x*y)**3]) c = CosetTable(f, [x*y]) c.scan_and_fill(0, x*y) assert c.table == [[1, None, None, 1], [None, 0, 0, None]] assert c.p == [0, 1] assert c.n == 2 assert c.omega == [0, 1] c.scan_and_fill(0, x**2) assert c.table == [[1, 1, None, 1], [0, 0, 0, None]] assert c.p == [0, 1] assert c.n == 2 assert c.omega == [0, 1] c.scan_and_fill(0, y**3) assert c.table == [[1, 1, 2, 1], [0, 0, 0, 2], [None, None, 1, 0]] assert c.p == [0, 1, 2] assert c.n == 3 assert c.omega == [0, 1, 2] c.scan_and_fill(0, (x*y)**3) assert c.table == [[1, 1, 2, 1], [0, 0, 0, 2], [None, None, 1, 0]] assert c.p == [0, 1, 2] assert c.n == 3 assert c.omega == [0, 1, 2] c.scan_and_fill(1, x**2) assert c.table == [[1, 1, 2, 1], [0, 0, 0, 2], [None, None, 1, 0]] assert c.p == [0, 1, 2] assert c.n == 3 assert c.omega == [0, 1, 2] c.scan_and_fill(1, y**3) assert c.table == [[1, 1, 2, 1], [0, 0, 0, 2], [None, None, 1, 0]] assert c.p == [0, 1, 2] assert c.n == 3 assert c.omega == [0, 1, 2] c.scan_and_fill(1, (x*y)**3) assert c.table == [[1, 1, 2, 1], [0, 0, 0, 2], [3, 4, 1, 0], [None, 2, 4, None], [2, None, None, 3]] assert c.p == [0, 1, 2, 3, 4] assert c.n == 5 assert c.omega == [0, 1, 2, 3, 4] c.scan_and_fill(2, x**2) assert c.table == [[1, 1, 2, 1], [0, 0, 0, 2], [3, 3, 1, 0], [2, 2, 3, 3], [2, None, None, 3]] assert c.p == [0, 1, 2, 3, 3] assert c.n == 4 assert c.omega == [0, 1, 2, 3] @slow def test_coset_enumeration(): # this test function contains the combined tests for the two strategies # i.e. HLT and Felsch strategies. # Example 5.1 from [1] F, x, y = free_group("x, y") f = FpGroup(F, [x**3, y**3, x**-1*y**-1*x*y]) C_r = coset_enumeration_r(f, [x]) C_r.compress(); C_r.standardize() C_c = coset_enumeration_c(f, [x]) C_c.compress(); C_c.standardize() table1 = [[0, 0, 1, 2], [1, 1, 2, 0], [2, 2, 0, 1]] assert C_r.table == table1 assert C_c.table == table1 # E1 from [2] Pg. 474 F, r, s, t = free_group("r, s, t") E1 = FpGroup(F, [t**-1*r*t*r**-2, r**-1*s*r*s**-2, s**-1*t*s*t**-2]) C_r = coset_enumeration_r(E1, []) C_r.compress() C_c = coset_enumeration_c(E1, []) C_c.compress() table2 = [[0, 0, 0, 0, 0, 0]] assert C_r.table == table2 # test for issue #11449 assert C_c.table == table2 # Cox group from [2] Pg. 474 F, a, b = free_group("a, b") Cox = FpGroup(F, [a**6, b**6, (a*b)**2, (a**2*b**2)**2, (a**3*b**3)**5]) C_r = coset_enumeration_r(Cox, [a]) C_r.compress(); C_r.standardize() C_c = coset_enumeration_c(Cox, [a]) C_c.compress(); C_c.standardize() table3 = [[0, 0, 1, 2], [2, 3, 4, 0], [5, 1, 0, 6], [1, 7, 8, 9], [9, 10, 11, 1], [12, 2, 9, 13], [14, 9, 2, 11], [3, 12, 15, 16], [16, 17, 18, 3], [6, 4, 3, 5], [4, 19, 20, 21], [21, 22, 6, 4], [7, 5, 23, 24], [25, 23, 5, 18], [19, 6, 22, 26], [24, 27, 28, 7], [29, 8, 7, 30], [8, 31, 32, 33], [33, 34, 13, 8], [10, 14, 35, 35], [35, 36, 37, 10], [30, 11, 10, 29], [11, 38, 39, 14], [13, 39, 38, 12], [40, 15, 12, 41], [42, 13, 34, 43], [44, 35, 14, 45], [15, 46, 47, 34], [34, 48, 49, 15], [50, 16, 21, 51], [52, 21, 16, 49], [17, 50, 53, 54], [54, 55, 56, 17], [41, 18, 17, 40], [18, 28, 27, 25], [26, 20, 19, 19], [20, 57, 58, 59], [59, 60, 51, 20], [22, 52, 61, 23], [23, 62, 63, 22], [64, 24, 33, 65], [48, 33, 24, 61], [62, 25, 54, 66], [67, 54, 25, 68], [57, 26, 59, 69], [70, 59, 26, 63], [27, 64, 71, 72], [72, 73, 68, 27], [28, 41, 74, 75], [75, 76, 30, 28], [31, 29, 77, 78], [79, 77, 29, 37], [38, 30, 76, 80], [78, 81, 82, 31], [43, 32, 31, 42], [32, 83, 84, 85], [85, 86, 65, 32], [36, 44, 87, 88], [88, 89, 90, 36], [45, 37, 36, 44], [37, 82, 81, 79], [80, 74, 41, 38], [39, 42, 91, 92], [92, 93, 45, 39], [46, 40, 94, 95], [96, 94, 40, 56], [97, 91, 42, 82], [83, 43, 98, 99], [100, 98, 43, 47], [101, 87, 44, 90], [82, 45, 93, 97], [95, 102, 103, 46], [104, 47, 46, 105], [47, 106, 107, 100], [61, 108, 109, 48], [105, 49, 48, 104], [49, 110, 111, 52], [51, 111, 110, 50], [112, 53, 50, 113], [114, 51, 60, 115], [116, 61, 52, 117], [53, 118, 119, 60], [60, 70, 66, 53], [55, 67, 120, 121], [121, 122, 123, 55], [113, 56, 55, 112], [56, 103, 102, 96], [69, 124, 125, 57], [115, 58, 57, 114], [58, 126, 127, 128], [128, 128, 69, 58], [66, 129, 130, 62], [117, 63, 62, 116], [63, 125, 124, 70], [65, 109, 108, 64], [131, 71, 64, 132], [133, 65, 86, 134], [135, 66, 70, 136], [68, 130, 129, 67], [137, 120, 67, 138], [132, 68, 73, 131], [139, 69, 128, 140], [71, 141, 142, 86], [86, 143, 144, 71], [145, 72, 75, 146], [147, 75, 72, 144], [73, 145, 148, 120], [120, 149, 150, 73], [74, 151, 152, 94], [94, 153, 146, 74], [76, 147, 154, 77], [77, 155, 156, 76], [157, 78, 85, 158], [143, 85, 78, 154], [155, 79, 88, 159], [160, 88, 79, 161], [151, 80, 92, 162], [163, 92, 80, 156], [81, 157, 164, 165], [165, 166, 161, 81], [99, 107, 106, 83], [134, 84, 83, 133], [84, 167, 168, 169], [169, 170, 158, 84], [87, 171, 172, 93], [93, 163, 159, 87], [89, 160, 173, 174], [174, 175, 176, 89], [90, 90, 89, 101], [91, 177, 178, 98], [98, 179, 162, 91], [180, 95, 100, 181], [179, 100, 95, 152], [153, 96, 121, 148], [182, 121, 96, 183], [177, 97, 165, 184], [185, 165, 97, 172], [186, 99, 169, 187], [188, 169, 99, 178], [171, 101, 174, 189], [190, 174, 101, 176], [102, 180, 191, 192], [192, 193, 183, 102], [103, 113, 194, 195], [195, 196, 105, 103], [106, 104, 197, 198], [199, 197, 104, 109], [110, 105, 196, 200], [198, 201, 133, 106], [107, 186, 202, 203], [203, 204, 181, 107], [108, 116, 205, 206], [206, 207, 132, 108], [109, 133, 201, 199], [200, 194, 113, 110], [111, 114, 208, 209], [209, 210, 117, 111], [118, 112, 211, 212], [213, 211, 112, 123], [214, 208, 114, 125], [126, 115, 215, 216], [217, 215, 115, 119], [218, 205, 116, 130], [125, 117, 210, 214], [212, 219, 220, 118], [136, 119, 118, 135], [119, 221, 222, 217], [122, 182, 223, 224], [224, 225, 226, 122], [138, 123, 122, 137], [123, 220, 219, 213], [124, 139, 227, 228], [228, 229, 136, 124], [216, 222, 221, 126], [140, 127, 126, 139], [127, 230, 231, 232], [232, 233, 140, 127], [129, 135, 234, 235], [235, 236, 138, 129], [130, 132, 207, 218], [141, 131, 237, 238], [239, 237, 131, 150], [167, 134, 240, 241], [242, 240, 134, 142], [243, 234, 135, 220], [221, 136, 229, 244], [149, 137, 245, 246], [247, 245, 137, 226], [220, 138, 236, 243], [244, 227, 139, 221], [230, 140, 233, 248], [238, 249, 250, 141], [251, 142, 141, 252], [142, 253, 254, 242], [154, 255, 256, 143], [252, 144, 143, 251], [144, 257, 258, 147], [146, 258, 257, 145], [259, 148, 145, 260], [261, 146, 153, 262], [263, 154, 147, 264], [148, 265, 266, 153], [246, 267, 268, 149], [260, 150, 149, 259], [150, 250, 249, 239], [162, 269, 270, 151], [262, 152, 151, 261], [152, 271, 272, 179], [159, 273, 274, 155], [264, 156, 155, 263], [156, 270, 269, 163], [158, 256, 255, 157], [275, 164, 157, 276], [277, 158, 170, 278], [279, 159, 163, 280], [161, 274, 273, 160], [281, 173, 160, 282], [276, 161, 166, 275], [283, 162, 179, 284], [164, 285, 286, 170], [170, 188, 184, 164], [166, 185, 189, 173], [173, 287, 288, 166], [241, 254, 253, 167], [278, 168, 167, 277], [168, 289, 290, 291], [291, 292, 187, 168], [189, 293, 294, 171], [280, 172, 171, 279], [172, 295, 296, 185], [175, 190, 297, 297], [297, 298, 299, 175], [282, 176, 175, 281], [176, 294, 293, 190], [184, 296, 295, 177], [284, 178, 177, 283], [178, 300, 301, 188], [181, 272, 271, 180], [302, 191, 180, 303], [304, 181, 204, 305], [183, 266, 265, 182], [306, 223, 182, 307], [303, 183, 193, 302], [308, 184, 188, 309], [310, 189, 185, 311], [187, 301, 300, 186], [305, 202, 186, 304], [312, 187, 292, 313], [314, 297, 190, 315], [191, 316, 317, 204], [204, 318, 319, 191], [320, 192, 195, 321], [322, 195, 192, 319], [193, 320, 323, 223], [223, 324, 325, 193], [194, 326, 327, 211], [211, 328, 321, 194], [196, 322, 329, 197], [197, 330, 331, 196], [332, 198, 203, 333], [318, 203, 198, 329], [330, 199, 206, 334], [335, 206, 199, 336], [326, 200, 209, 337], [338, 209, 200, 331], [201, 332, 339, 240], [240, 340, 336, 201], [202, 341, 342, 292], [292, 343, 333, 202], [205, 344, 345, 210], [210, 338, 334, 205], [207, 335, 346, 237], [237, 347, 348, 207], [208, 349, 350, 215], [215, 351, 337, 208], [352, 212, 217, 353], [351, 217, 212, 327], [328, 213, 224, 323], [354, 224, 213, 355], [349, 214, 228, 356], [357, 228, 214, 345], [358, 216, 232, 359], [360, 232, 216, 350], [344, 218, 235, 361], [362, 235, 218, 348], [219, 352, 363, 364], [364, 365, 355, 219], [222, 358, 366, 367], [367, 368, 353, 222], [225, 354, 369, 370], [370, 371, 372, 225], [307, 226, 225, 306], [226, 268, 267, 247], [227, 373, 374, 233], [233, 360, 356, 227], [229, 357, 361, 234], [234, 375, 376, 229], [248, 231, 230, 230], [231, 377, 378, 379], [379, 380, 359, 231], [236, 362, 381, 245], [245, 382, 383, 236], [384, 238, 242, 385], [340, 242, 238, 346], [347, 239, 246, 381], [386, 246, 239, 387], [388, 241, 291, 389], [343, 291, 241, 339], [375, 243, 364, 390], [391, 364, 243, 383], [373, 244, 367, 392], [393, 367, 244, 376], [382, 247, 370, 394], [395, 370, 247, 396], [377, 248, 379, 397], [398, 379, 248, 374], [249, 384, 399, 400], [400, 401, 387, 249], [250, 260, 402, 403], [403, 404, 252, 250], [253, 251, 405, 406], [407, 405, 251, 256], [257, 252, 404, 408], [406, 409, 277, 253], [254, 388, 410, 411], [411, 412, 385, 254], [255, 263, 413, 414], [414, 415, 276, 255], [256, 277, 409, 407], [408, 402, 260, 257], [258, 261, 416, 417], [417, 418, 264, 258], [265, 259, 419, 420], [421, 419, 259, 268], [422, 416, 261, 270], [271, 262, 423, 424], [425, 423, 262, 266], [426, 413, 263, 274], [270, 264, 418, 422], [420, 427, 307, 265], [266, 303, 428, 425], [267, 386, 429, 430], [430, 431, 396, 267], [268, 307, 427, 421], [269, 283, 432, 433], [433, 434, 280, 269], [424, 428, 303, 271], [272, 304, 435, 436], [436, 437, 284, 272], [273, 279, 438, 439], [439, 440, 282, 273], [274, 276, 415, 426], [285, 275, 441, 442], [443, 441, 275, 288], [289, 278, 444, 445], [446, 444, 278, 286], [447, 438, 279, 294], [295, 280, 434, 448], [287, 281, 449, 450], [451, 449, 281, 299], [294, 282, 440, 447], [448, 432, 283, 295], [300, 284, 437, 452], [442, 453, 454, 285], [309, 286, 285, 308], [286, 455, 456, 446], [450, 457, 458, 287], [311, 288, 287, 310], [288, 454, 453, 443], [445, 456, 455, 289], [313, 290, 289, 312], [290, 459, 460, 461], [461, 462, 389, 290], [293, 310, 463, 464], [464, 465, 315, 293], [296, 308, 466, 467], [467, 468, 311, 296], [298, 314, 469, 470], [470, 471, 472, 298], [315, 299, 298, 314], [299, 458, 457, 451], [452, 435, 304, 300], [301, 312, 473, 474], [474, 475, 309, 301], [316, 302, 476, 477], [478, 476, 302, 325], [341, 305, 479, 480], [481, 479, 305, 317], [324, 306, 482, 483], [484, 482, 306, 372], [485, 466, 308, 454], [455, 309, 475, 486], [487, 463, 310, 458], [454, 311, 468, 485], [486, 473, 312, 455], [459, 313, 488, 489], [490, 488, 313, 342], [491, 469, 314, 472], [458, 315, 465, 487], [477, 492, 485, 316], [463, 317, 316, 468], [317, 487, 493, 481], [329, 447, 464, 318], [468, 319, 318, 463], [319, 467, 448, 322], [321, 448, 467, 320], [475, 323, 320, 466], [432, 321, 328, 437], [438, 329, 322, 434], [323, 474, 452, 328], [483, 494, 486, 324], [466, 325, 324, 475], [325, 485, 492, 478], [337, 422, 433, 326], [437, 327, 326, 432], [327, 436, 424, 351], [334, 426, 439, 330], [434, 331, 330, 438], [331, 433, 422, 338], [333, 464, 447, 332], [449, 339, 332, 440], [465, 333, 343, 469], [413, 334, 338, 418], [336, 439, 426, 335], [441, 346, 335, 415], [440, 336, 340, 449], [416, 337, 351, 423], [339, 451, 470, 343], [346, 443, 450, 340], [480, 493, 487, 341], [469, 342, 341, 465], [342, 491, 495, 490], [361, 407, 414, 344], [418, 345, 344, 413], [345, 417, 408, 357], [381, 446, 442, 347], [415, 348, 347, 441], [348, 414, 407, 362], [356, 408, 417, 349], [423, 350, 349, 416], [350, 425, 420, 360], [353, 424, 436, 352], [479, 363, 352, 435], [428, 353, 368, 476], [355, 452, 474, 354], [488, 369, 354, 473], [435, 355, 365, 479], [402, 356, 360, 419], [405, 361, 357, 404], [359, 420, 425, 358], [476, 366, 358, 428], [427, 359, 380, 482], [444, 381, 362, 409], [363, 481, 477, 368], [368, 393, 390, 363], [365, 391, 394, 369], [369, 490, 480, 365], [366, 478, 483, 380], [380, 398, 392, 366], [371, 395, 496, 497], [497, 498, 489, 371], [473, 372, 371, 488], [372, 486, 494, 484], [392, 400, 403, 373], [419, 374, 373, 402], [374, 421, 430, 398], [390, 411, 406, 375], [404, 376, 375, 405], [376, 403, 400, 393], [397, 430, 421, 377], [482, 378, 377, 427], [378, 484, 497, 499], [499, 499, 397, 378], [394, 461, 445, 382], [409, 383, 382, 444], [383, 406, 411, 391], [385, 450, 443, 384], [492, 399, 384, 453], [457, 385, 412, 493], [387, 442, 446, 386], [494, 429, 386, 456], [453, 387, 401, 492], [389, 470, 451, 388], [493, 410, 388, 457], [471, 389, 462, 495], [412, 390, 393, 399], [462, 394, 391, 410], [401, 392, 398, 429], [396, 445, 461, 395], [498, 496, 395, 460], [456, 396, 431, 494], [431, 397, 499, 496], [399, 477, 481, 412], [429, 483, 478, 401], [410, 480, 490, 462], [496, 497, 484, 431], [489, 495, 491, 459], [495, 460, 459, 471], [460, 489, 498, 498], [472, 472, 471, 491]] assert C_r.table == table3 assert C_c.table == table3 # Group denoted by B2,4 from [2] Pg. 474 F, a, b = free_group("a, b") B_2_4 = FpGroup(F, [a**4, b**4, (a*b)**4, (a**-1*b)**4, (a**2*b)**4, \ (a*b**2)**4, (a**2*b**2)**4, (a**-1*b*a*b)**4, (a*b**-1*a*b)**4]) C_r = coset_enumeration_r(B_2_4, [a]) C_c = coset_enumeration_c(B_2_4, [a]) index_r = 0 for i in range(len(C_r.p)): if C_r.p[i] == i: index_r += 1 assert index_r == 1024 index_c = 0 for i in range(len(C_c.p)): if C_c.p[i] == i: index_c += 1 assert index_c == 1024 # trivial Macdonald group G(2,2) from [2] Pg. 480 M = FpGroup(F, [b**-1*a**-1*b*a*b**-1*a*b*a**-2, a**-1*b**-1*a*b*a**-1*b*a*b**-2]) C_r = coset_enumeration_r(M, [a]) C_r.compress(); C_r.standardize() C_c = coset_enumeration_c(M, [a]) C_c.compress(); C_c.standardize() table4 = [[0, 0, 0, 0]] assert C_r.table == table4 assert C_c.table == table4 def test_look_ahead(): # Section 3.2 [Test Example] Example (d) from [2] F, a, b, c = free_group("a, b, c") f = FpGroup(F, [a**11, b**5, c**4, (a*c)**3, b**2*c**-1*b**-1*c, a**4*b**-1*a**-1*b]) H = [c, b, c**2] table0 = [[1, 2, 0, 0, 0, 0], [3, 0, 4, 5, 6, 7], [0, 8, 9, 10, 11, 12], [5, 1, 10, 13, 14, 15], [16, 5, 16, 1, 17, 18], [4, 3, 1, 8, 19, 20], [12, 21, 22, 23, 24, 1], [25, 26, 27, 28, 1, 24], [2, 10, 5, 16, 22, 28], [10, 13, 13, 2, 29, 30]] CosetTable.max_stack_size = 10 C_c = coset_enumeration_c(f, H) C_c.compress(); C_c.standardize() assert C_c.table[: 10] == table0 def test_modified_methods(): ''' Tests for modified coset table methods. Example 5.7 from [1] Holt, D., Eick, B., O'Brien "Handbook of Computational Group Theory". ''' F, x, y = free_group("x, y") f = FpGroup(F, [x**3, y**5, (x*y)**2]) H = [x*y, x**-1*y**-1*x*y*x] C = CosetTable(f, H) C.modified_define(0, x) identity = C._grp.identity a_0 = C._grp.generators[0] a_1 = C._grp.generators[1] assert C.P == [[identity, None, None, None], [None, identity, None, None]] assert C.table == [[1, None, None, None], [None, 0, None, None]] C.modified_define(1, x) assert C.table == [[1, None, None, None], [2, 0, None, None], [None, 1, None, None]] assert C.P == [[identity, None, None, None], [identity, identity, None, None], [None, identity, None, None]] C.modified_scan(0, x**3, C._grp.identity, fill=False) assert C.P == [[identity, identity, None, None], [identity, identity, None, None], [identity, identity, None, None]] assert C.table == [[1, 2, None, None], [2, 0, None, None], [0, 1, None, None]] C.modified_scan(0, x*y, C._grp.generators[0], fill=False) assert C.P == [[identity, identity, None, a_0**-1], [identity, identity, a_0, None], [identity, identity, None, None]] assert C.table == [[1, 2, None, 1], [2, 0, 0, None], [0, 1, None, None]] C.modified_define(2, y**-1) assert C.table == [[1, 2, None, 1], [2, 0, 0, None], [0, 1, None, 3], [None, None, 2, None]] assert C.P == [[identity, identity, None, a_0**-1], [identity, identity, a_0, None], [identity, identity, None, identity], [None, None, identity, None]] C.modified_scan(0, x**-1*y**-1*x*y*x, C._grp.generators[1]) assert C.table == [[1, 2, None, 1], [2, 0, 0, None], [0, 1, None, 3], [3, 3, 2, None]] assert C.P == [[identity, identity, None, a_0**-1], [identity, identity, a_0, None], [identity, identity, None, identity], [a_1, a_1**-1, identity, None]] C.modified_scan(2, (x*y)**2, C._grp.identity) assert C.table == [[1, 2, 3, 1], [2, 0, 0, None], [0, 1, None, 3], [3, 3, 2, 0]] assert C.P == [[identity, identity, a_1**-1, a_0**-1], [identity, identity, a_0, None], [identity, identity, None, identity], [a_1, a_1**-1, identity, a_1]] C.modified_define(2, y) assert C.table == [[1, 2, 3, 1], [2, 0, 0, None], [0, 1, 4, 3], [3, 3, 2, 0], [None, None, None, 2]] assert C.P == [[identity, identity, a_1**-1, a_0**-1], [identity, identity, a_0, None], [identity, identity, identity, identity], [a_1, a_1**-1, identity, a_1], [None, None, None, identity]] C.modified_scan(0, y**5, C._grp.identity) assert C.table == [[1, 2, 3, 1], [2, 0, 0, 4], [0, 1, 4, 3], [3, 3, 2, 0], [None, None, 1, 2]] assert C.P == [[identity, identity, a_1**-1, a_0**-1], [identity, identity, a_0, a_0*a_1**-1], [identity, identity, identity, identity], [a_1, a_1**-1, identity, a_1], [None, None, a_1*a_0**-1, identity]] C.modified_scan(1, (x*y)**2, C._grp.identity) assert C.table == [[1, 2, 3, 1], [2, 0, 0, 4], [0, 1, 4, 3], [3, 3, 2, 0], [4, 4, 1, 2]] assert C.P == [[identity, identity, a_1**-1, a_0**-1], [identity, identity, a_0, a_0*a_1**-1], [identity, identity, identity, identity], [a_1, a_1**-1, identity, a_1], [a_0*a_1**-1, a_1*a_0**-1, a_1*a_0**-1, identity]] # Modified coset enumeration test f = FpGroup(F, [x**3, y**3, x**-1*y**-1*x*y]) C = coset_enumeration_r(f, [x]) C_m = modified_coset_enumeration_r(f, [x]) assert C_m.table == C.table