{"id":1265,"date":"2021-12-05T23:10:57","date_gmt":"2021-12-05T15:10:57","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=1265"},"modified":"2025-02-26T11:05:16","modified_gmt":"2025-02-26T03:05:16","slug":"free_group_mark","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2021\/12\/05\/free_group_mark\/","title":{"rendered":"\u81ea\u7531\u7fa4\u6ce8\u8bb0"},"content":{"rendered":"<p><script type=\"text\/javascript\" async src=\"https:\/\/www.caiqinyi.cn\/wp-content\/MathJax\/MathJax.js?config=TeX-AMS_CHTML\">\n<\/script><br \/>\n<script type=\"text\/x-mathjax-config\">\n    MathJax.Hub.Config({\n        tex2jax: {inlineMath: [['$','$']]},\n        TeX: {equationNumbers: {autoNumber: [\"AMS\"], useLabelIds: true}},\n        \"HTML-CSS\": {linebreaks: {automatic: true}},\n        SVG: {linebreaks: {automatic: true}}\n    });\n<\/script><\/p>\n<p>\u53c8\u662f\u5f00\u59cb\u5468\u516d\u52a0\u73ed\u7684\u4e00\u5468, \u54ce, \u8fd8\u662f\u8981\u5728\u4e1a\u4f59\u65f6\u95f4\u575a\u6301\u5b66\u5b8c\u4ee3\u6570\u62d3\u6251\u543c\u543c~ \u76ee\u524d\u5b66\u4e60\u5230\u6709\u9650\u8868\u51fa\u7fa4\u4e00\u8282, \u8bf4\u5b9e\u8bdd, \u770b\u5230\u4ea4\u6362\u5316\u7cfb\u6570\u77e9\u9635\u90e8\u5206\u65f6, \u6211\u5df2\u7ecf\u6709\u70b9\u61f5\u903c\u4e86\u2026\u2026 \u6240\u4ee5\u6700\u8fd1\u4e24\u5468\u4ee5\u6765\u91cd\u5fc3\u4e3b\u8981\u653e\u5728\u4e86\u5e84\u6653\u6ce2\u8001\u5e08\u7684\u4ee3\u6570\u62d3\u6251\u7684\u89c6\u9891\u5b66\u4e60\u4e0a, \u5e0c\u671b\u80fd\u52a0\u6df1\u81ea\u7531\u79ef\u4e0e\u751f\u6210\u5143\u7b49\u77e5\u8bc6\u70b9\u7684\u7406\u89e3. \u672c\u6587\u65e8\u5728\u7b80\u5355\u8bb0\u5f55\u4e00\u4e0b\u81ea\u7531\u7fa4\u76f8\u5173\u7684\u77e5\u8bc6\u70b9.<\/p>\n<p><!--more--><\/p>\n<p><strong>\u53c2\u8003\u6750\u6599<\/strong><br \/>\n1. <a href=\"https:\/\/zhuanlan.zhihu.com\/p\/34684100\">\u81ea\u7531\u7fa4<\/a><br \/>\n2. <a href=\"https:\/\/zhuanlan.zhihu.com\/p\/164799373\">5. \u7f6e\u6362\u7fa4\u3001\u5355\u7fa4\u3001\u53ef\u89e3\u7fa4\u3001\u81ea\u540c\u6784\u7fa4\u3001\u81ea\u7531\u7fa4<\/a><\/p>\n<p>\u901a\u8fc7\u5b57\u7684\u7b49\u4ef7\u7c7b\u53bb\u5b9a\u4e49\u81ea\u7531\u7fa4, \u8fd9\u662f\u4e2a\u4eba\u6bd4\u8f83\u559c\u6b22\u540c\u65f6\u89c9\u5f97\u4e5f\u6bd4\u8f83\u5bb9\u6613\u7406\u89e3\u7684\u65b9\u5f0f. \u5728\u5b9a\u4e49\u81ea\u7531\u7fa4\u4e4b\u524d, \u6211\u4eec\u5148\u5b9a\u4e49\u5b57\u7684\u7b49\u4ef7\u7c7b: \u5982\u679c\u4e24\u4e2a\u5b57\u901a\u8fc7\u7ea6\u5316\u64cd\u4f5c\u80fd\u591f\u5f97\u5230\u540c\u4e00\u4e2a\u65e2\u7ea6\u6587\u5b57, \u5219\u79f0\u8fd9\u4e24\u4e2a\u5b57\u662f\u7b49\u4ef7\u7684. \u8fd9\u6837\u4e00\u6765, \u6211\u4eec\u4fbf\u53ef\u4ee5\u5b9a\u4e49\u81ea\u7531\u7fa4.<\/p>\n<p><strong>\u5b9a\u4e49<\/strong> \u8bbe$F(S)$\u4e3a\u96c6\u5408$S \\cup S^{-1}$\u4e0a\u5b57\u7684\u7b49\u4ef7\u7c7b\u96c6\u5408, \u5b9a\u4e49$F(S)$\u4e0a\u7684\u4e58\u6cd5\u4e3a$[u][v] $$ = [uv]$, \u5219$F(S)$\u5728\u6b64\u4e58\u6cd5\u5b9a\u4e49\u4e0b\u6784\u6210\u4e00\u4e2a\u7fa4, \u79f0$F(S)$\u4e3a\u5b9a\u4e49\u5728$S$\u4e0a\u7684\u81ea\u7531\u7fa4.<br \/>\n$\\\\$ \u7531\u4e0a\u8ff0\u5b9a\u4e49\u53ef\u77e5, \u82e5\u60f3\u8981\u628a\u4efb\u610f\u4e00\u4e2a\u96c6\u5408\u6539\u9020\u6210\u81ea\u7531\u7fa4, \u5148\u8981\u53d6\u8be5\u96c6\u5408\u7684\u9006\u96c6(\u5373\u8be5\u96c6\u5408\u7684\u6240\u6709\u5143\u7d20\u5728\u4e58\u6cd5\u8fd0\u7b97\u4e0b\u7684\u9006\u5143\u6784\u6210\u7684\u96c6\u5408), \u518d\u6784\u9020\u65e2\u7ea6\u6587\u5b57, \u6700\u540e\u8fdb\u884c\u4e58\u6cd5\u8fd0\u7b97\u5373\u53ef.<br \/>\n$\\\\$ \u81ea\u7531\u7fa4\u7684\u610f\u4e49\u5728\u4e8e\u5b83\u7684\u5546\u7fa4\u53ef\u4ee5\u8868\u8fbe\u4efb\u4e00\u7fa4. \u4e5f\u5c31\u662f\u8bf4, \u6211\u4eec\u53ef\u4ee5\u5728\u81ea\u7531\u7fa4\u4e0a\u6dfb\u52a0\u5404\u79cd\u5404\u6837\u7684\u5173\u7cfb, \u4f7f\u5f97\u5b83\u4eec\u80fd\u53d8\u6210\u5404\u79cd\u5404\u6837\u7684\u7fa4, \u4f8b\u5982\u5faa\u73af\u7fa4, Abel\u7fa4, \u5176\u7406\u8bba\u4fdd\u8bc1\u662f\u5982\u4e0b\u5b9a\u7406,<\/p>\n<blockquote><p>\u4efb\u610f\u4e00\u7fa4$G$\u90fd\u662f\u67d0\u4e2a\u81ea\u7531\u7fa4\u7684\u540c\u6001\u50cf.<\/p><\/blockquote>\n<p>\u7531\u4e0a\u8ff0\u5b9a\u7406\u53ef\u77e5, \u540c\u6001(\u7531\u540c\u6001\u57fa\u672c\u5b9a\u7406\u53ef\u77e5\u76f8\u5f53\u4e8e\u5546\u7fa4) \u5c31\u662f\u6dfb\u52a0\u4e00\u79cd\u7b49\u4ef7\u5173\u7cfb, \u4ece\u800c\u6211\u4eec\u80fd\u591f\u8fdb\u4e00\u6b65\u5206\u7c7b. \u6211\u4eec\u5bf9\u4e00\u4e2a\u96c6\u5408\u6dfb\u52a0\u5173\u7cfb\u540e\u53ef\u4ee5\u5f97\u5230\u4e00\u4e2a\u81ea\u7531\u7fa4, \u518d\u6dfb\u52a0\u5173\u7cfb, \u53c8\u53ef\u4ee5\u5f97\u5230\u4efb\u610f\u4e00\u4e2a\u7fa4.<\/p>\n<p>PS:<br \/>\n$\\\\$ 1. \u4e00\u5b9a\u8981\u5b66\u4f1a\u4ece\u5173\u7cfb\u7684\u89d2\u5ea6\u6765\u89e3\u91ca\u4ee3\u6570\u5b66\u4e2d\u7684\u5f88\u591a\u6982\u5ff5, \u6709\u4e9b\u770b\u8d77\u6765\u4e0d\u6e05\u4e0d\u695a\u7684, \u901a\u8fc7\u89e3\u8bfb\u5173\u7cfb, \u90fd\u80fd\u53d8\u5f97\u5f88\u6e05\u6670.<br \/>\n$\\\\$ 2. \u81ea\u7531\u7fa4\u6dfb\u52a0\u5173\u7cfb\u540e\u53d8\u6210\u4efb\u610f\u4e00\u4e2a\u7fa4, \u8fd9\u53e5\u8bdd\u5176\u5b9e\u7ed9\u51fa\u4e86\u4e00\u4e2a\u4efb\u610f\u7fa4\u7684\u6807\u51c6\u5199\u6cd5, \u6211\u4eec\u53ea\u8981\u7ed9\u51fa\u4e00\u4e2a\u81ea\u7531\u7fa4, \u7136\u540e\u6ce8\u660e\u5b83\u6dfb\u52a0\u4e86\u4ec0\u4e48\u6837\u7684\u5173\u7cfb\u5c31\u591f\u4e86.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u53c8\u662f\u5f00\u59cb\u5468\u516d\u52a0\u73ed\u7684\u4e00\u5468, \u54ce, \u8fd8\u662f\u8981\u5728\u4e1a\u4f59\u65f6\u95f4\u575a\u6301\u5b66\u5b8c\u4ee3\u6570\u62d3\u6251\u543c\u543c~ \u76ee\u524d\u5b66\u4e60\u5230\u6709\u9650\u8868\u51fa\u7fa4\u4e00\u8282, \u8bf4\u5b9e\u8bdd,  &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/12\/05\/free_group_mark\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u81ea\u7531\u7fa4\u6ce8\u8bb0<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[15],"tags":[],"_links":{"self":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1265"}],"collection":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/comments?post=1265"}],"version-history":[{"count":13,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1265\/revisions"}],"predecessor-version":[{"id":3615,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1265\/revisions\/3615"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=1265"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=1265"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=1265"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}