{"id":2254,"date":"2012-08-06T18:45:49","date_gmt":"2012-08-06T10:45:49","guid":{"rendered":"http:\/\/vanabel.sinaapp.com\/?p=2254"},"modified":"2012-08-06T18:45:49","modified_gmt":"2012-08-06T10:45:49","slug":"%e9%ad%94%e6%96%b9%e8%bf%98%e5%8e%9f%e7%ae%97%e6%b3%95%e5%88%9d%e6%8e%a2","status":"publish","type":"post","link":"https:\/\/lttt.vanabel.cn\/?p=2254","title":{"rendered":"\u9b54\u65b9\u8fd8\u539f(\u673a\u68b0)\u7b97\u6cd5\u521d\u63a2"},"content":{"rendered":"<p>\u9996\u5148, \u6211\u60f3\u8bf4\u660e\u4e0b\u9898\u76ee. \u7ecf\u8fc73\u5929\u7684\u4e0d\u65ad\u8bd5\u9a8c\u548c\u601d\u8003, \u6211\u5b8c\u5168\u5b9e\u73b0\u4e86\u4e8c\u9636\u9b54\u65b9\u7684\u8fd8\u539f\u7684\u673a\u68b0\u7b97\u6cd5. \u6362\u8a00\u4e4b, \u4f60\u53ef\u4ee5\u4e00\u6b65\u4e00\u6b65\u5730\u6309\u7167\u6211\u8bf4\u7684\u7b97\u6cd5\u8fd8\u539f\u4e8c\u9636\u9b54\u65b9. \u57fa\u672c\u60f3\u6cd5\u662f\u8fd9\u6837\u7684, \u9996\u5148\u5c06\u6bcf\u4e2a\u9762\u7f16\u53f7(1,2,&#8230;,24), \u90a3\u4e48\u73b0\u5728\u4efb\u4f55\u4e00\u4e2a\u8f6c\u52a8\u90fd\u5bf9\u5e94\u7740(1,2,&#8230;24)\u7684\u4e00\u4e2a\u7f6e\u6362, \u7279\u522b\u5730, \u6211\u4eec\u53ef\u4ee5\u5199\u51fa6\u4e2a\u57fa\u672c\u64cd\u4f5c\u7684\u7f6e\u6362\u8868\u793a; \u5176\u6b21, \u5229\u7528\u8fd9\u516d\u4e2a\u7f6e\u6362\u8868\u793a\u751f\u62102\u9636\u9b54\u65b9\u7fa4, \u5e76\u5229\u7528GAP\u8f6f\u4ef6\u5efa\u7acb\u8be5\u7fa4\u548c\u75316\u4e2a\u81ea\u7531\u5143\u751f\u6210\u7684\u81ea\u7531\u7fa4\u7684\u540c\u6001; \u6700\u540e\u901a\u8fc7\u8f93\u5165\u9b54\u65b9\u7684\u7ed9\u5b9a\u72b6\u6001(\u5373\u7ecf\u8fc7\u82e5\u5e72\u8f6c\u52a8\u540e\u5f97\u5230\u7684\u9b54\u65b9)\u5728\u8be5\u81ea\u7531\u7fa4\u4e0b\u7684\u751f\u6210\u5143\u800c\u5f97\u5230\u5177\u4f53\u7684\u8fd8\u539f\u6b65\u9aa4. \u4e0b\u9762\u8ba9\u6211\u8be6\u7ec6\u53d9\u8ff0\u4e4b, \u5e76\u4e0d\u65f6\u63d2\u5165\u6211\u9677\u5165\u7684\u8bef\u533a. \u4f60\u4f1a\u770b\u5230\u8fd8\u6709\u8bb8\u591a\u672a\u89e3\u51b3\u7684\u95ee\u9898(\u6700\u77ed\u8981\u591a\u5c11\u6b65\u8fd8\u539f, \u5982\u4f55\u9009\u53d6\u66f4\u4f18\u7684\u751f\u6210\u5143, \u5982\u4f55\u53d1\u73b0\u5feb\u901f\u516c\u5f0f\u7b49), \u4f46\u662f\u5e76\u4e0d\u662f\u8bf4\u4e0d\u80fd\u901a\u8fc7\u6211\u7684\u65b9\u6cd5\u8fd8\u539f.<!--more--><\/p>\n<h3>\u9b54\u65b9\u6bcf\u4e2a\u9762\u7684\u7f16\u53f7\u65b9\u6cd5<\/h3>\n<p>\u4e3a\u4e86\u7b80\u5355\u8d77\u89c1, \u6211\u4eec\u628a\u4e00\u6761\u68f1\u6b63\u5bf9\u7740\u81ea\u5df1, \u800c\u4e14\u7f16\u53f7\u4ece\u4e0a, \u5230\u524d, \u518d\u5230\u53f3, \u521d\u59cb\u90fd\u662f\u4ee5\u4e0a\u9762\u7684\u90a3\u4e2a\u9876\u89d2\u5757(\u5373\u7f16\u53f7\u4e3a1,5,9\u7684\u5757)\u5f00\u59cb\u7684, \u7136\u540e\u5bf9\u524d\u540e\u53f3\u4e09\u4e2a\u9762\u4ee5\u9006\u65f6\u9488\u7f16\u53f7, \u6700\u540e\u5bf9\u9762\u7684\u7f16\u53f7\u7528\u4e00\u4e2a\u5c0f\u62ec\u53f7\u533a\u5206.<br \/>\n\u6700\u7ec8\u7684\u6548\u679c\u5982\u56fe\u6240\u793a:<br \/>\n<a href=\"http:\/\/i.imgur.com\/0Oobb.png\"><img decoding=\"async\" class=\"aligncenter\" title=\"color coding\" src=\"http:\/\/i.imgur.com\/0Oobb.png\" alt=\"\" width=\"50%\" \/><\/a><\/p>\n<h3>\u57fa\u672c\u64cd\u4f5c\u7684\u7f6e\u6362\u8868\u793a<\/h3>\n<p>\u6b63\u5982\u5f15\u8a00\u4e2d\u63d0\u5230\u7684, \u4e00\u65e6\u7f16\u53f7\u4e86\u5404\u4e2a\u9762, \u90a3\u4e48\u6bcf\u4e00\u4e2a\u57fa\u672c\u7684\u64cd\u4f5c\u90fd\u53ef\u4ee5\u7528\u4e00\u4e2a\u7f6e\u6362\u6765\u8868\u793a. \u6211\u4eec\u77e5\u9053\u9b54\u65b9\u7684\u57fa\u672c\u64cd\u4f5c\u6709\u5bf9\u4e0a(u), \u4e0b(d), \u524d(f), \u540e(b), \u53f3(r), \u5de6(l)\u5404\u4e2a\u9762\u7684\u65cb\u8f6c90\u5ea6. \u800c\u4e14\u53ea\u9700\u8981\u9006\u65f6\u9488\u65cb\u8f6c\u5373\u53ef(\u987a\u65f6\u9488\u53ef\u4ee5\u901a\u8fc7\u9006\u65f6\u9488\u4e09\u6b21\u5f97\u5230); \u53ef\u80fd\u6709\u7684\u4eba\u53ef\u80fd\u4f1a\u8ba4\u4e3a\u53ea\u9700\u7531\u4e0a,\u4e0b,\u524d\u8fd9\u51e0\u79cd\u64cd\u4f5c\u5b8c\u6210, \u6211\u4eec\u5f85\u4f1a\u4f1a\u770b\u5230(\u8bc1\u660e)\u8fd9\u5176\u5b9e\u662f\u4e0d\u5bf9\u7684.<br \/>\n\u90a3\u4e48\u5177\u4f53\u800c\u8a00, \u5728\u4e0a\u8ff0\u7f16\u53f7\u4e0b, \u5404\u4e2a\u57fa\u672c\u64cd\u4f5c\u7684\u7f6e\u6362\u8868\u793a\u662f\u4ec0\u4e48\u5462? \u4e3e\u4f8b\u6765\u8bf4, u\u8fd9\u4e2a\u5bf9\u9876\u9762\u7684\u9006\u65f6\u9488\u65cb\u8f6c90\u5ea6\u628a1\u8fd9\u4e2a\u9762\u53d8\u4e3a2, \u4e3a\u4e86\u7b80\u4fbf\u5c06\u5176\u8bb0\u4e3a1-&gt;2, \u540c\u65f6\u67092-&gt;3, 3-&gt;4, 4-&gt;1; \u5728\u7fa4\u8bba\u4e2d, \u6211\u4eec\u4e60\u60ef\u8fd9\u4e2a&#8221;\u5faa\u73af&#8221;\u8bb0\u4e3a(1,2,3,4), \u5373\u5b83\u8868\u793a\u4e00\u4e2a\u53d8\u6362, \u5c061-&gt;2-&gt;3-&gt;4-&gt;1. \u5229\u7528\u8fd9\u4e2a\u8bb0\u53f7, \u6211\u4eec\u53ef\u4ee5\u77e5\u9053u=(1,2,3,4)(5,12,18,13)(9,17,16,6)\u8fd9\u4e09\u4e2a\u5faa\u73af\u7684\u4e58\u79ef; \u7c7b\u4f3c\u7684\u53ef\u5f97\u5230\u5176\u4ed6\u57fa\u672c\u64cd\u4f5c\u7684\u7f6e\u6362\u8868\u793a. \u6211\u4eec\u5217\u4e3e\u5982\u4e0b:<\/p>\n<pre lang=\"c\" line=\"N\" file=\"\" colla=\"+\">u=(1,2,3,4)(5,12,18,13)(9,17,16,6);\nd=(21,22,23,24)(7,10,20,15)(8,11,19,14);\nf=(5,6,7,8)(1,13,24,10)(4,14,21,9);\nb=(17,18,19,20)(2,16,23,11)(3,15,22,12);\nr=(9,10,11,12)(5,21,20,2)(8,22,17,1);\nl=(13,14,15,16)(3,6,24,19)(4,7,23,18);<\/pre>\n<p>\u5229\u7528\u8be5\u8868\u793a, \u6211\u4eec\u5c31\u53ef\u4ee5\u5f97\u5230\u7531\u5b83\u4eec\u751f\u6210\u7684\u7fa4: RubikGroup2. \u8fdb\u800c\u5229\u7528\u6570\u5b66\u8f6f\u4ef6\u53ef\u4ee5\u8ba1\u7b97\u51fa\u5b83\u7684\u9636\u662f:88179840.<\/p>\n<p>\u4e0b\u9762\u662f\u5b83\u4eec\u5728\u5e38\u7528\u6570\u5b66\u8f6f\u4ef6\u4e0b\u7684\u4ee3\u7801:<\/p>\n<ul>\n<li>mathematica:\n<pre lang=\"c\" line=\"N\" file=\"mathematica\" colla=\"+\">u = Cycles[{{1, 2, 3, 4}, {5, 12, 18, 13}, {9, 17, 16, 6}}];\nd = Cycles[{{21, 22, 23, 24}, {7, 10, 20, 15}, {8, 11, 19, 14}}];\nf = Cycles[{{5, 6, 7, 8}, {1, 13, 24, 10}, {4, 14, 21, 9}}];\nb = Cycles[{{17, 18, 19, 20}, {2, 16, 23, 11}, {3, 15, 22, 12}}];\nr = Cycles[{{9, 10, 11, 12}, {5, 21, 20, 2}, {8, 22, 17, 1}}];\nl = Cycles[{{13, 14, 15, 16}, {3, 6, 24, 19}, {4, 7, 23, 18}}];\nRubikGroup2 = PermutationGroup[{r, d, f, b, r, l}];\nGroupOrder[RubikGroup2]<\/pre>\n<\/li>\n<li>maple:\n<pre lang=\"c\" line=\"N\" file=\"maple\" colla=\"+\">&gt; with(group);\n&gt; pg := permgroup(24, {b = [[17, 18, 19, 20], [2, 16, 23, 11], [3, 15, 22, 12]], d = [[21, 22, 23, 24], [7, 10, 20, 15], [8, 11, 19, 14]], f = [[5, 6, 7, 8], [1, 13, 24, 10], [4, 14, 21, 9]], l = [[13, 14, 15, 16], [3, 6, 24, 19], [4, 7, 23, 18]], r = [[9, 10, 11, 12], [5, 21, 20, 2], [8, 22, 17, 1]], u = [[1, 2, 3, 4], [5, 12, 18, 13], [9, 17, 16, 6]]});\n&gt; ordg := grouporder(pg);<\/pre>\n<\/li>\n<li>GAP:\n<pre lang=\"c\" line=\"N\" file=\"GAP\" colla=\"+\">RubikGroup2:=Group((1,2,3,4)(5,12,18,13)(9,17,16,6),(21,22,23,24)(7,10,20,15)(8,11,19,14),\n(5,6,7,8)(1,13,24,10)(4,14,21,9),(17,18,19,20)(2,16,23,11)(3,15,22,12),\n(9,10,11,12)(5,21,20,2)(8,22,17,1),(13,14,15,16)(3,6,24,19)(4,7,23,18));\nSize(RubikGroup2);<\/pre>\n<\/li>\n<\/ul>\n<p>\u4e8e\u662f, \u53ef\u4ee5\u5229\u7528\u4e0a\u9762\u7684\u4ee3\u7801\u9a8c\u8bc1\u4ec5\u6709u, f, r \u4e09\u79cd\u57fa\u672c\u64cd\u4f5c\u751f\u6210\u7684\u7fa4\u7684\u9636\u662f3674160, \u4e8e\u662f\u4e0e\u539f\u6765\u76842\u9636\u9b54\u65b9\u7fa4\u4e0d\u540c\u6784.<br \/>\n\u8fd9\u91cc, \u6211\u53ea\u7ed9\u51famaple\u4e0b\u7684\u9a8c\u8bc1\u4ee3\u7801:<\/p>\n<pre lang=\"c\" line=\"N\" file=\"maple\" colla=\"+\">&gt; sg := permgroup(24, {f = [[5, 6, 7, 8], [1, 13, 24, 10], [4, 14, 21, 9]], r = [[9, 10, 11, 12], [5, 21, 20, 2], [8, 22, 17, 1]], u = [[1, 2, 3, 4], [5, 12, 18, 13], [9, 17, 16, 6]]});\n&gt; ordsg := grouporder(sg);<\/pre>\n<h3>\u901a\u8fc7\u989c\u8272\u5bfb\u6c42\u4ece\u521d\u59cb\u72b6\u6001\u53d8\u6362\u5230\u7ed9\u5b9a\u72b6\u6001\u7684\u7f6e\u6362<\/h3>\n<p>\u4e3a\u6b64, \u6211\u4eec\u9700\u8981\u8bc6\u522b\u9b54\u65b9\u4e0a\u7684\u989c\u8272, \u4e0d\u59a8\u5047\u8bbe\u8fd8\u539f\u72b6\u6001\u4e0b\u9b54\u65b9\u4e0a(u), \u524d(f), \u53f3(r)\u7684\u989c\u8272\u5206\u522b\u662f\u7ea2(R), \u7eff(G), \u9ed1(B), \u800c\u4e14\u5176\u5bf9\u7acb\u9762\u7684\u989c\u8272\u7528\u5c0f\u5199\u5b57\u6bcd\u8868\u793a. \u5982\u56fe\u6240\u793a:<br \/>\n<a href=\"http:\/\/i.imgur.com\/6woxH.png\"><img decoding=\"async\" class=\"aligncenter\" title=\"numbering with faces\" src=\"http:\/\/i.imgur.com\/6woxH.png\" alt=\"\" width=\"50%\" \/><\/a><br \/>\n\u90a3\u4e48, \u4ece\u7406\u8bba\u4e0a\u8bf4, \u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u5bf9\u7ed9\u5b9a\u72b6\u6001\u989c\u8272\u7684\u7f16\u53f7\u5e8f\u5217\u5f97\u51fa\u8be5\u72b6\u6001\u7684\u7f6e\u6362\u8868\u793a, \u4f8b\u5982:\u521d\u59cb\u72b6\u6001\u4e0b\u5404\u4e2a\u9762\u7684\u989c\u8272\u6309\u7167\u7f16\u53f7\u987a\u6b21\u662f:R,R,R,R,G,G,G,G,B,B,B,B,b,b,b,b,g,g,g,g,r,r,r,r; \u800c\u7ed9\u5b9a\u72b6\u6001\u7684\u989c\u8272\u6309\u7167\u7f16\u53f7\u987a\u5e8f\u4f9d\u6b21\u662f:R,R,R,R,G,G,b,G,B,B,r,B,b,g,r,b,g,g,g,b,r,G,B,r; \u90a3\u4e48\u6211\u4eec\u8981\u6c42\u8ba1\u7b97\u51fa\u4ece\u521d\u59cb\u72b6\u6001(\u5404\u9762\u540c\u8272)\u5230\u7ed9\u5b9a\u72b6\u6001\u7684\u7f6e\u6362(\u5b83\u662f\u4e00\u7cfb\u5217u,d,f,b,r,l\u7684\u590d\u5408). \u6ce8\u610f\u5230\u8fd9\u65f6\u6211\u4eec\u5e94\u8be5\u5728\u9b54\u65b9\u7fa4\u4e2d\u53bb\u627e(\u6bd4\u8f83\u96be), \u4e0d\u80fd\u572824\u9636\u7f6e\u6362\u7fa4\u4e2d\u53bb\u627e(\u5bb9\u6613). \u5176\u5b9e\u901a\u8fc7\u6240\u8c13\u7684\u5f3a\u751f\u6210\u96c6, \u6211\u4eec\u53ef\u4ee5\u5f88\u5bb9\u6613\u627e\u5230\u8be5\u7f6e\u6362, \u4f46\u662f\u6211\u8fd8\u6ca1\u5b66\u4f1a. (\u6ce8\u610f\u5f88\u591a\u8f6f\u4ef6\u90fd\u53ef\u4ee5\u7b97\u51fa\u7a33\u5b9a\u94fe\u4ee5\u53ca\u57fa, \u4e8e\u662f\u8fd9\u79cd\u65b9\u6cd5\u662f\u53ef\u884c\u7684). \u6211\u8fd9\u91cc\u5148\u7ed9\u51fa\u8ba1\u7b97\u5f3a\u751f\u6210\u96c6\u4ee5\u53ca\u7a33\u5b9a\u94fe\u7684mathematica\u4ee3\u7801:<\/p>\n<pre lang=\"c\" line=\"N\" file=\"mathematica\" colla=\"+\">Stabchain = GroupStabilizerChain[RubikGroup2];\nStabchain \/. gr_PermutationGroup :&gt; GroupOrder[gr] \/\/ Column<\/pre>\n<p>\u53ef\u89c1, \u9b54\u65b9\u7fa4\u7684\u4e00\u4e2a\u57fa\u662f{1, 2, 3, 7, 4, 8, 11}, \u4e0e\u4e4b\u76f8\u5bf9\u7684\u5f3a\u751f\u6210\u5143\u662f:<br \/>\n{(11, 20, 22)(15, 19, 23),<br \/>\n(11, 23)(15, 22)(19, 20),<br \/>\n(8, 21, 10)(11, 22, 20)(15, 19, 23),<br \/>\n(8, 20, 19)(10, 22, 15)(11, 23, 21),<br \/>\n(4, 15, 6, 23, 13, 19)(8, 22, 21, 20, 10, 11),<br \/>\n(7, 10, 20, 15)(8, 11, 19, 14)(21, 22, 23, 24),<br \/>\n(3, 6, 24, 19)(4, 7, 23, 18)(13, 14, 15, 16),<br \/>\n(2, 16, 23, 11)(3, 15, 22, 12)(17, 18, 19, 20),<br \/>\n(1, 13, 24, 10)(4, 14, 21, 9)(5, 6, 7, 8)}<br \/>\n\u5f97\u5230\u5f3a\u751f\u6210\u5143\u7684mathematica\u4ee3\u7801:<\/p>\n<pre lang=\"c\" line=\"N\" file=\"mathematica\" colla=\"+\">strongGS = GroupGenerators[Stabchain[[1, -1]]]<\/pre>\n<p>\u65e0\u8bba\u5982\u4f55, \u4f60\u53ef\u4ee5\u901a\u8fc7\u76f4\u63a5\u89c2\u5bdf, \u5f97\u51fa\u6211\u4e3e\u7684\u90a3\u4e2a\u4f8b\u5b50\u7684\u7f6e\u6362\u662f\u5c06:<br \/>\nR1, R2, R3, R4,<br \/>\nG1, G2, G3, G4,<br \/>\nB1, B2, B3, B4,<br \/>\nb1, b2, b3, b4,<br \/>\ng1, g2, g3, g4,<br \/>\nr1, r2, r3, r4<br \/>\n\u53d8\u4e3a<br \/>\nR1, R2, R3, R4,<br \/>\nG1, G2, b3, G4,<br \/>\nB1, B2, r4, B4,<br \/>\nb1, g3, r2, b4,<br \/>\ng1, g2, g4, b2,<br \/>\nr1, G3, B3, r3.<br \/>\n\u4e8e\u662f\u5f97\u5230\u8be5\u72b6\u6001\u7684\u7f6e\u6362(rep)\u662f:<br \/>\n(7, 22, 15)(11, 23, 24)(14, 20, 19),<br \/>\n\u53ef\u4ee5\u5229\u7528mathematica\u5982\u4e0b\u4ee3\u7801\u83b7\u5f97:<\/p>\n<pre lang=\"c\" line=\"N\" file=\"mathematica\" colla=\"+\">Clear[b, r];\niniorder = {\n   R1, R2, R3, R4,\n   G1, G2, G3, G4,\n   B1, B2, B3, B4,\n   b1, b2, b3, b4,\n   g1, g2, g3, g4,\n   r1, r2, r3, r4};\nendorder = {\n   R1, R2, R3, R4,\n   G1, G2, b3, G4,\n   B1, B2, r4, B4,\n   b1, g3, r2, b4,\n   g1, g2, g4, b2,\n   r1, G3, B3, r3};\nrep = FindPermutation[iniorder, endorder]\nGroupElementQ[RubikGroup2, rep]<\/pre>\n<p>\u6700\u540e\u4e00\u884c\u662f\u6d4b\u8bd5\u662f\u4e0d\u662f\u9b54\u65b9\u7fa4\u4e2d\u7684\u4e00\u4e2a\u5143\u7d20(\u5c31\u662f\u8fd9\u4e2a\u6d4b\u8bd5\u4f7f\u5f97\u6211\u53d1\u73b0\u6211\u539f\u6765\u7684\u7f6e\u6362\u662f\u5728$S_{24}$\u91cc\u9762\u505a\u7684);<\/p>\n<p>\u81f3\u6b64, \u6211\u4eec\u5df2\u7ecf\u628a\u95ee\u9898\u8f6c\u5316\u4e3a\u628arep\u8fd9\u4e2a\u72b6\u6001\u7f6e\u6362\u7528u,d,f,b,r,l\u53ca\u5b83\u4eec\u5404\u81ea\u7684\u9006\u5177\u4f53\u5730\u8868\u793a\u51fa\u6765. \u4e3a\u6b64, \u6211\u4eec\u9700\u8981\u501f\u52a9\u81ea\u7531\u7fa4\u4ee5\u53ca\u6709\u9650\u7fa4\u8bba\u8ba1\u7b97\u8f6f\u4ef6GAP.<\/p>\n<h3>GAP\u4e0b\u5bf9rep\u7684\u5206\u89e3<\/h3>\n<h4>\u81ea\u7531\u7fa4\u7684\u6784\u9020<\/h4>\n<p>\u5efa\u7acb\u4ee5&#8221;u&#8221;,&#8221;d&#8221;,&#8221;f&#8221;,&#8221;b&#8221;,&#8221;r&#8221;,&#8221;l&#8221;\u4e3a&#8221;\u5b57\u6bcd&#8221;\u7684\u81ea\u7531\u7fa4,<br \/>\nGAP\u4ee3\u7801:<\/p>\n<pre lang=\"c\" line=\"N\" file=\"GAP\" colla=\"+\">freeg:=FreeGroup(\"u\",\"d\",\"f\",\"b\",\"r\",\"l\");;<\/pre>\n<h4>\u7fa4\u540c\u6001\u7684\u5efa\u7acb<\/h4>\n<p>GAP\u4ee3\u7801:<\/p>\n<pre lang=\"c\" line=\"N\" file=\"GAP\" colla=\"+\">hom:=GroupHomomorphismByImages(freeg,RubikGroup2,GeneratorsOfGroup(freeg),GeneratorsOfGroup(RubikGroup2));<\/pre>\n<h4>\u5c06\u72b6\u6001\u7f6e\u6362\u5206\u89e3\u4e3a\u57fa\u672c\u64cd\u4f5c(\u4e00\u4e2a\u5177\u4f53\u7684\u5143\u7d20\u5206\u89e3\u4e3a\u751f\u6210\u5143\u7684\u4e58\u79ef)<\/h4>\n<p>\u4ee5\u6211\u4e3e\u4f8b\u7684\u90a3\u4e2a\u72b6\u6001\u7f6e\u6362\u4e3a\u4f8b,<br \/>\nGAP\u4ee3\u7801:<\/p>\n<pre lang=\"c\" line=\"N\" file=\"GAP\" colla=\"+\">rep:=(7, 22, 15)(11, 23, 24)(14, 20, 19);<\/pre>\n<p>\u6700\u540e\u628a\u8fd9\u4e2a\u5143\u7d20(rep)\u5206\u89e3\u4e3a\u57fa\u672c\u64cd\u4f5c\u7684\u4e58\u79ef:<br \/>\nGAP\u4ee3\u7801:<\/p>\n<pre lang=\"c\" line=\"N\" file=\"GAP\" colla=\"+\">w:=PreImagesRepresentative(hom,rep);<\/pre>\n<p>GAP\u8f93\u51fa\u7ed3\u679c\u4e3a:<\/p>\n<pre lang=\"c\" line=\"N\" file=\"GAP\" colla=\"+\">d*b*u^-1*r*b^2*f^-1*r*f*r*u^-1*r^-1<\/pre>\n<p>\u8fd9\u8868\u660e, \u5bf9\u9b54\u65b9\u5b9e\u884crep\u7684\u9006\u5c31\u53ef\u4ee5\u8fd8\u539f\u9b54\u65b9\u4e86. \u5373\u4f9d\u6b21\u8fdb\u884cr,u,r\u9006,f\u9006,b,b,r\u9006,u,b\u9006,d\u9006.<br \/>\n\u5176\u4e2d, \u5982\u524d\u6240\u5047\u8bbe(\u6ce8\u610f\u9b54\u65b9\u7684\u653e\u7f6e\u5e94\u548c\u7b2c\u4e00\u4e2a\u56fe\u4e00\u81f4), r\u8868\u793a\u53f3\u65b9\u7684\u9762\u9006\u65f6\u9488\u65cb\u8f6c90\u5ea6, u\u8868\u793a\u4e0a\u65b9\u7684\u9762\u9006\u8f6c90\u5ea6, \u540c\u7406\u6709\u5176\u4ed6\u7684\u5b57\u6bcd, \u800cr\u9006\u5f53\u7136\u5c31\u662f\u8868\u793a\u53f3\u65b9\u7684\u9762\u987a\u65f6\u9488\u65cb\u8f6c90\u5ea6, \u5176\u4ed6\u7c7b\u4f3c\u5f97\u5230.<\/p>\n<h3>\u4e00\u4e9b\u601d\u8003<\/h3>\n<p>\u672c\u63a2\u7d22\u5b9e\u73b0\u4e86\u5c06\u4e00\u4e2a2\u9636\u9b54\u65b9\u7684\u4efb\u4e00\u72b6\u6001\u8fd8\u539f\u7684\u6b65\u9aa4(\u5206\u89e3\u4e3a\u57fa\u672c\u64cd\u4f5c\u53ca\u5176\u9006\u7684\u4e58\u79ef), \u4f46\u662f\u8fd8\u6709\u5982\u4e0b\u51e0\u70b9\u6ca1\u6709\u89e3\u51b3:<\/p>\n<ol>\n<li>\u672c\u7b97\u6cd5\u5f88\u53ef\u80fd\u4e0d\u662f\u6700\u4f18\u7684, \u5373\u4e0d\u662f\u628a\u7ed9\u5b9a\u72b6\u6001\u8fd8\u539f\u4e3a\u521d\u59cb\u72b6\u6001\u6240\u9700\u7684\u6700\u5c11\u7684\u6b65\u9aa4;<\/li>\n<li>\u5982\u4f55\u6539\u9020\u53d8\u751f\u6210\u5143, \u4f7f\u5176\u80fd\u5728\u5927\u591a\u6570\u60c5\u5f62\u4e0b\u66f4\u5feb\u7684\u5b8c\u6210\u8fd8\u539f<\/li>\n<li>\u5bf9RubkiGroup2\u7684\u9636\u8fdb\u884c\u7d20\u56e0\u6570\u5206\u89e3\u53ef\u4ee5\u77e5\u9053\u5b83\u67095\u9636\u5143\u548c7\u9636\u5143, \u662f\u5426\u6709\u8fd9\u4e24\u4e2a\u5143\u6269\u5145\u6210\u751f\u6210\u5143\u4f1a\u6bd4\u539f\u6765\u76846\u79cd\u57fa\u672c\u64cd\u4f5c\u66f4\u4f18.<\/li>\n<li>\u5982\u4f55\u7528\u5f3a\u751f\u6210\u5143\u6765\u83b7\u5f97\u72b6\u6001\u7f6e\u6362\u7684\u8868\u793a?<\/li>\n<\/ol>\n<h3>\u53c2\u8003\u6587\u732e<\/h3>\n<ol>\n<li><a title=\"Factorization in finite Groups\" href=\"www.math.colostate.edu\/~hulpke\/talks\/decomp.pdf\">Factorization in finite Groups<\/a><\/li>\n<\/ol>\n<h3>\u9644\u8bb0<\/h3>\n<ul>\n<li>\u6587\u4e2d\u90a3\u4e2a\u5f3a\u751f\u6210\u5143\u7684\u5206\u89e3:<\/li>\n<pre lang=\"c\" line=\"\" file=\"GAP\" colla=\"+\">(11, 20, 22)(15, 19, 23)=f*b^-1*l^-1*f^-2*r^2*f*u^-1*f^-1*r*f*u^-1*r*u^-1*f*u^-1*f^-1*u*r^-1\n(11, 23)(15, 22)(19, 20)=r^-1*f^-1*u*l*b*r*d^-1*u*b^-1*u*r*u^-1*r^-1*d^-1*u*r^-1*d*r^-1*b*r*b^-1*u*f^-1*r^-2*f*b\n(8, 21, 10)(11, 22, 20)(15, 19, 23)=r*b*u^-1*r*b^2*f^-1*r*f*r^-1*f*u*f^-1*u^-1*f^-1*u*b*u^-1*f*u*b^-1*u^-1*r*u^-1*f*u*f^-1*u*r^-1*u*f^-1*r^-1*f*u^-1\n(8, 20, 19)(10, 22, 15)(11, 23, 21)=u*f^-1*r*f*u^-1*r*u^-1*f*u^-1*f^-1*u*r^-1*b*u^-1*b^-1*r*u*r^-1*u*f^-1*r^-2*f*b^-2*r^-2*b^-1\n(4, 15, 6, 23, 13, 19)(8, 22, 21, 20, 10, 11)=u*r*u^-1*r^-1*b*u*b^-1*f^-1*r^-1*f*u^-1*f*l*f^-1*u*r\n(7, 10, 20, 15)(8, 11, 19, 14)(21, 22, 23, 24)=d\n(3, 6, 24, 19)(4, 7, 23, 18)(13, 14, 15, 16)=l\n(2, 16, 23, 11)(3, 15, 22, 12)(17, 18, 19, 20)=b\n(1, 13, 24, 10)(4, 14, 21, 9)(5, 6, 7, 8)=f<\/pre>\n<li>\u6587\u4e2d\u7684\u90a3\u4e2a2\u9636\u9b54\u65b9\u7fa4RubikGroup2\u53ef\u4ee5\u7531\u4e24\u4e2a\u751f\u6210\u5143\u751f\u6210:\n<pre lang=\"c\" line=\"\" file=\"GAP\" colla=\"+\">(2,24,21,13,19,16,20)(3,11,12,7,10,6,23)(4,15,18,22,17,14,8)=r^2*u^-1*r^-1*f*u^-1*f^-1*r*u*r^-1*f^-1*r^-1*u*r*d^-1*b*r^-1*b^-1*u^-1*b^-2*u^-2*r^-1*u^-1*d\n(1,10,9,21,5,8)(2,3,23,13,11)(4,20,17,16,19)(6,22,12, 18,15)(7,14,24) =r^-1*f^-1*u*l*b*r*d^-1*u*b^-1*u*r*u^-1*r^-1*u*r*u^-1*r^-1*b*u*b^-1*f^-1*r*f*u^-1*r*u^-1*r^-1*f*d^-1*f^-1*l*r^-1*u*b^\n-1*r^-1*f*d^-1<\/pre>\n<\/li>\n<li>\u6211\u4eec\u4e5f\u53ef\u4ee5\u5f97\u5230\u7fa4\u7684\u62bd\u8c61\u8868\u793a(GAP\u4ee3\u7801):\n<pre lang=\"c\" line=\"\" file=\"GAP\" colla=\"+\">iso:=IsomorphismFpGroup(RubikGroup2);<\/pre>\n<p>\u5f97\u5230\u4e00\u4e2a\u540c\u6784\u8868\u793a(GAP\u8f93\u51fa):<\/p>\n<pre lang=\"c\" line=\"\" file=\"GAP\" colla=\"+\">[ (11,19)(15,20)(22,23), (1,2,3,4,7,8,15)(5,12,18,13,14,21,19)(6,24,10,23,9,17,16), (1,2,3,4,7,11,15)(5,12,18,13,14,\n    22,19)(6,24,20,23,9,17,16), (11,22,20)(15,23,19), (1,9,5)(15,19,23), (2,17,12)(15,19,23), (3,18,16)(15,23,19),\n  (4,13,6)(15,23,19), (7,24,14)(15,19,23), (8,21,10)(15,23,19) ] -&gt; [ F1, F2, F3, F4, F5, F6, F7, F8, F9, F10 ]<\/pre>\n<p>\u5229\u7528\u8be5\u540c\u6784\u8868\u793a, \u6211\u4eec\u53ef\u4ee5\u5f97\u52302\u9636\u9b54\u65b9\u7fa4\u7684\u62bd\u8c61\u5173\u7cfb(\u7528F1&#8230;F10\u6765\u8868\u793a), GAP\u4ee3\u7801\u4e3a:<\/p>\n<pre lang=\"c\" line=\"\" file=\"GAP\" colla=\"+\"> fp:=Image(iso);;\n RelatorsOfFpGroup(fp);<\/pre>\n<p>GAP\u8f93\u51fa\u7684\u62bd\u8c61\u5143\u4e4b\u95f4\u7684\u5173\u7cfb\u4e3a:<\/p>\n<pre lang=\"c\" line=\"\" file=\"GAP\" colla=\"+\">[F1^2,\nF1^-1*F2*F1*F2^-1*F3^-1*F2*F10^-2*F4^-1,\nF1^-1*F3*F1*F3*F2^-1*F3^-2*F2^2*F3*F2^-2*F3*F2^-1*F9^-2*F4^-1,\n\nF2^7*F3^-7,\nF2^7*F3^-1*F2^-1*F3^-1*F2^-1*F3^-1*F2^-1*F3^-1*F2^-1,\nF2^-1*F3^-1*F2*F3*F2^-1*F3^-1*F2*F3,\n\nF2^-2*F3^-1*F2*F3^2*F2^-2*F3^-1*F2*F3^2,\nF2^-3*F3^-1*F2*F3^3*F2^-3*F3^-1*F2*F3^3,\nF1^-1*F4*F1*F4^-2,\n\nF1^-1*F5*F1*F5^-1*F4^-1,\nF1^-1*F6*F1*F6^-1*F4^-1,\nF1^-1*F7*F1*F7^-1*F4^-2,\nF1^-1*F8*F1*F8^-1*F4^-2,\n\nF1^-1*F9*F1*F9^-1*F4^-1,\nF1^-1*F10*F1*F10^-1*F4^-2,\nF2^-1*F4*F2*F5^-1*F4^-1,\nF2^-1*F5*F2*F6^-1*F5^-2,\n\nF2^-1*F6*F2*F7^-2*F5^-2,\nF2^-1*F7*F2*F8^-1*F5^-1,\nF2^-1*F8*F2*F9^-2*F5^-1,\nF2^-1*F9*F2*F10^-2*F5^-2,\n\nF2^-1*F10*F2*F5^-1,\nF3^-1*F4*F3*F5^-1,\nF3^-1*F5*F3*F6^-1*F5^-2,\nF3^-1*F6*F3*F7^-2*F5^-2,\nF3^-1*F7*F3*F8^-1*F5^-1,\n\nF3^-1*F8*F3*F9^-2*F5^-1,\nF3^-1*F9*F3*F5^-2*F4^-2,\nF3^-1*F10*F3*F10^-1*F5^-1,\nF4^3,\nF5^3,\nF5^-1*F4^-1*F5*F4,\nF6^3,\n\nF6^-1*F4^-1*F6*F4,\nF6^-1*F5^-1*F6*F5,\nF7^3,\nF7^-1*F4^-1*F7*F4,\nF7^-1*F5^-1*F7*F5,\nF7^-1*F6^-1*F7*F6,\nF8^3,\n\nF8^-1*F4^-1*F8*F4,\nF8^-1*F5^-1*F8*F5,\nF8^-1*F6^-1*F8*F6,\nF8^-1*F7^-1*F8*F7,\nF9^3,\nF9^-1*F4^-1*F9*F4,\n\nF9^-1*F5^-1*F9*F5,\nF9^-1*F6^-1*F9*F6,\nF9^-1*F7^-1*F9*F7,\nF9^-1*F8^-1*F9*F8,\nF10^3,\nF10^-1*F4^-1*F10*F4,\n\nF10^-1*F5^-1*F10*F5,\nF10^-1*F6^-1*F10*F6,\nF10^-1*F7^-1*F10*F7,\nF10^-1*F8^-1*F10*F8,\nF10^-1*F9^-1*F10*F9]<\/pre>\n<p>\u6bcf\u4e00\u4e2a\u90fd\u662f\u6052\u7b49\u5143, \u4f8b\u5982\u7b2c\u4e00\u4e2aF1^2=id,\u662f\u5bb9\u6613\u770b\u51fa\u6765\u7684.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>\u9996\u5148, \u6211\u60f3\u8bf4\u660e\u4e0b\u9898\u76ee. \u7ecf\u8fc73\u5929\u7684\u4e0d\u65ad\u8bd5\u9a8c\u548c\u601d\u8003, \u6211\u5b8c\u5168\u5b9e\u73b0\u4e86\u4e8c\u9636\u9b54\u65b9\u7684\u8fd8\u539f &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"more-link\" href=\"https:\/\/lttt.vanabel.cn\/?p=2254\"> <span class=\"screen-reader-text\">\u9b54\u65b9\u8fd8\u539f(\u673a\u68b0)\u7b97\u6cd5\u521d\u63a2<\/span> \u9605\u8bfb\u66f4\u591a &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[482,611,197,758,305,385,403],"class_list":["post-2254","post","type-post","status-publish","format-standard","hentry","category-net","tag-gap","tag-rubik","tag-197","tag-758","tag-305","tag-385","tag-403"],"_links":{"self":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/2254","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2254"}],"version-history":[{"count":0,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/2254\/revisions"}],"wp:attachment":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2254"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2254"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2254"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}