{"id":1206,"date":"2021-10-09T11:32:20","date_gmt":"2021-10-09T03:32:20","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=1206"},"modified":"2022-10-19T10:18:39","modified_gmt":"2022-10-19T02:18:39","slug":"group_common_knowledge_review","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2021\/10\/09\/group_common_knowledge_review\/","title":{"rendered":"\u7fa4\u7684\u5e38\u7528\u77e5\u8bc6\u590d\u4e60"},"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>\u4eca\u5929\u7b97\u662f\u56fd\u5e86\u5047\u671f\u7684\u6700\u540e\u4e00\u5929\u4e86, \u867d\u7136\u660e\u5929\u4e5f\u4e0d\u7528\u4e0a\u73ed, \u4f46\u65f6\u95f4\u57fa\u672c\u90fd\u82b1\u5728\u8def\u4e0a\u4e86, \u6240\u4ee5\u6211\u5185\u5fc3\u4e5f\u5c31\u4e0d\u628a\u660e\u5929\u5f53\u5047\u671f\u4e86\u2026\u2026 \u54ce, \u53c8\u8981\u56de\u53bb996\u4e86, \u5185\u5fc3\u6709\u4e00\u4e22\u4e22\u4e0d\u723d, \u4f46\u8fd9\u4e5f\u662f\u65b0\u751f\u4ee3\u519c\u6c11\u5de5\u7684\u65e0\u5948\u5427QAQ \u4e0d\u8fc7\u4eca\u5e74\u7684\u8fd9\u4e2a\u56fd\u5e86\u5047\u671f\u7b97\u8d77\u6765\u4e5f\u670911\u5929\u4e86\u5462, \u4e5f\u8be5\u77e5\u8db3\u4e86\u54c8\u54c8~ \u672c\u5468\u4e3b\u8981\u590d\u4e60\u4e86\u5173\u4e8e\u7fa4\u7684\u5e38\u7528\u77e5\u8bc6, \u4ee5\u53ca\u5b66\u4e60\u4e86\u57fa\u672c\u7fa4\u7684\u6982\u5ff5. \u7fa4\u7684\u6982\u5ff5\u662f\u5728\u672c\u79d1\u65f6\u4fbf\u63a5\u89e6\u8fc7\u7684, \u6240\u4ee5\u672c\u6587\u4e5f\u4e0d\u518d\u8d58\u8ff0, \u53ea\u8bb0\u5f55\u4e00\u4e9b\u4e2a\u4eba\u611f\u89c9\u7406\u89e3\u5f97\u66f4\u4e3a\u6df1\u523b\u7684\u5173\u4e8e\u7fa4\u7684\u77e5\u8bc6\u70b9.<\/p>\n<p><!--more--><\/p>\n<p><strong>1. \u7fa4\u7684\u5b9a\u4e49<\/strong><\/p>\n<p>\u5982\u679c\u96c6\u5408$G$\u4ee5\u53ca\u6620\u5c04$\\mu : G \\times G \\to G$\u6ee1\u8db3:<br \/>\n$\\\\$ (1) \u7ed3\u5408\u5f8b: $\\forall a, b, c \\in G$,$$\\mu(\\mu(a, b), c) = \\mu(a, \\mu(b, c));$$(2) \u5b58\u5728\u5e7a\u5143$1_\\mu \\in G$, \u4f7f\u5f97$\\forall a \\in G$,$$\\mu(1_\\mu, a)  =\\mu(a, 1_\\mu) = a;$$(3) \u4efb\u53d6$a \\in G$, \u5b58\u5728\u9006\u5143$a^{-1}_\\mu \\in G$, \u4f7f\u5f97$$\\mu(a, a^{-1}_\\mu) = \\mu(a^{-1}_\\mu, a) = 1_\\mu,$$\u5219\u79f0$(G,\\mu)$\u4e3a\u4e00\u4e2a\u7fa4, \u79f0$\\mu$\u4e3a\u7fa4\u7684\u8fd0\u7b97.<br \/>\n$\\\\$ \u5bb9\u6613\u9a8c\u8bc1\u4e00\u4e2a\u7fa4$G$\u7684\u5e7a\u5143\u4e00\u5b9a\u662f\u552f\u4e00\u7684, \u800c\u4e14\u4efb\u53d6$a \\in G$, $a$\u7684\u9006\u5143\u4e5f\u4e00\u5b9a\u662f\u552f\u4e00\u7684. \u5728\u4e0d\u5f15\u8d77\u6df7\u6dc6\u7684\u60c5\u51b5\u4e0b, \u6211\u4eec\u4e5f\u628a\u7fa4\u7684\u8fd0\u7b97\u7b80\u79f0\u4e3a\u4e58\u6cd5, \u5e76\u628a$\\mu(a, b)$\u7b80\u8bb0\u4e3a$ab$, \u628a$1_\\mu$\u7b80\u8bb0\u4e3a1, \u628a$a^{-1}_\\mu$\u7b80\u8bb0\u4e3a$a^{-1}$. \u8fd9\u6837, \u5c31\u53ef\u4ee5\u628a\u5173\u4e8e\u7fa4\u7684\u8fd9\u4e09\u6761\u516c\u7406\u91cd\u5199\u6210\u4e0b\u8ff0\u66f4\u5bb9\u6613\u7406\u89e3\u548c\u8bb0\u5fc6\u7684\u65b9\u5f0f:<br \/>\n(1) \u4e58\u6cd5\u7ed3\u5408\u5f8b: $(ab)c = a(bc)$;<br \/>\n(2) \u5b58\u5728\u5e7a\u51431, \u4f7f\u5f97$1a = a1 = a$;<br \/>\n(3) \u6bcf\u4e2a$a$\u6709\u9006\u5143$a^{-1}$, \u4f7f\u5f97$aa^{-1} = a^{-1}a = 1$.<br \/>\n$\\\\$ \u73b0\u5728\u5f88\u591a\u6559\u6750\u90fd\u76f4\u63a5\u91c7\u7528\u4e86\u7b2c\u4e8c\u79cd\u88ab\u7b80\u5316\u540e\u7684\u5b9a\u4e49, \u6211\u4e2a\u4eba\u89c9\u5f97\u662f\u4e0d\u591f\u4e25\u8c28\u7684, \u5e94\u8be5\u628a\u6620\u5c04\u7684\u6982\u5ff5\u4e5f\u5f15\u5165\u5176\u4e2d, \u8fd9\u6837\u624d\u80fd\u66f4\u6df1\u523b\u5730\u7406\u89e3\u7fa4\u7684\u8fd0\u7b97. \u540c\u65f6, \u5f15\u5165\u6620\u5c04$\\mu:G \\times G \\to G$\u4e5f\u95f4\u63a5\u5730\u8bf4\u660e\u4e86\u7fa4\u7684\u8fd0\u7b97\u7684\u5c01\u95ed\u6027.<\/p>\n<p><strong>2. \u5176\u5b83\u6982\u5ff5<\/strong><\/p>\n<p>$\\cdot$ <strong>\u5e73\u51e1\u5b50\u7fa4\u4e0e\u5e73\u51e1\u7fa4:<\/strong> \u6bcf\u4e2a\u7fa4\u81f3\u5c11\u8981\u542b\u6709\u4e00\u4e2a\u5143\u7d20, \u5373\u5e7a\u51431, \u4e8e\u662f\u6bcf\u4e2a\u7fa4$G$\u7684\u5e7a\u5143\u90fd\u5355\u72ec\u6784\u6210$G$\u7684\u4e00\u4e2a\u5b50\u7fa4, \u79f0\u4e3a\u5176\u5e73\u51e1\u5b50\u7fa4. \u4e00\u4e2a\u7fa4\u4e2d\u5982\u679c\u9664\u4e86\u5e7a\u5143\u5916\u6ca1\u6709\u5176\u5b83\u5143\u7d20, \u5219\u79f0\u4e4b\u4e3a\u5e73\u51e1\u7fa4.<br \/>\n$\\\\$ $\\cdot$ <strong>\u6320\u5143\u4e0e\u81ea\u7531\u5143:<\/strong> \u82e5\u5b58\u5728\u6b63\u6574\u6570$n$, \u4f7f\u5f97$a^n = 1$, \u5219\u79f0$a$\u4e3a\u6320\u5143, \u5426\u5219\u79f0$a$\u4e3a\u81ea\u7531\u5143. \u8bbe$f:G \\to H$\u662f\u540c\u6001, \u5219$f(a^k) = f(a)^k$, \u8fd9\u8bf4\u660e\u540c\u6001\u628a\u6320\u5143\u4e00\u5b9a\u53d8\u6210\u6320\u5143(\u4f46\u662f\u53ef\u4ee5\u628a\u81ea\u7531\u5143\u53d8\u6210\u81ea\u7531\u5143\u6216\u6320\u5143). \u7279\u522b\u5730, \u5982\u679c\u6709\u540c\u6001$f: Z_n \\to Z$, \u5176\u4e2d, $Z_n = \\{ 0, 1, \\cdots, n &#8211; 1 \\}$, $Z_n$\u4e0a\u7684\u8fd0\u7b97\u4e0e\u6a21$n$\u7684\u8fd0\u7b97\u6709\u5173, \u5219$f$\u662f\u5e73\u51e1\u540c\u6001, \u56e0\u4e3a$Z_n$\u4e2d\u7684\u5143\u7d20\u5168\u90fd\u662f\u6320\u5143, \u800c$Z$\u4e2d\u7684\u975e0\u5143\u7d20\u5168\u90fd\u662f\u81ea\u7531\u5143.<br \/>\n$\\\\$ $\\cdot$ <strong>\u81ea\u7136\u540c\u6001:<\/strong> \u8bbe$H$\u662f$G$\u7684\u4e00\u4e2a\u6b63\u89c4\u5b50\u7fa4, \u8bb0\u6240\u6709$H$\u7684\u5de6\u966a\u96c6(\u4e5f\u7b49\u4e8e\u53f3\u966a\u96c6) \u6784\u6210\u7684\u96c6\u5408\u4e3a$G\/H$, \u5728\u5176\u4e0a\u5b9a\u4e49\u4e58\u6cd5\u8fd0\u7b97\u4e3a$$\\mu(aH, bH) = abH,$$\u5219$(G\/H, \\mu)$\u4e5f\u6784\u6210\u4e00\u4e2a\u7fa4, \u79f0\u4e3a$G$\u5173\u4e8e$H$\u7684\u5546\u7fa4.$$\\pi: G \\to G\/H, a \\mapsto aH$$\u662f\u7fa4\u7684\u540c\u6001, \u79f0\u4e3a\u8be5\u5546\u7fa4\u7684\u81ea\u7136\u540c\u6001.<\/p>\n<p><strong>3. \u540c\u6001\u57fa\u672c\u5b9a\u7406<\/strong><\/p>\n<p>\u8bbe$f:G \\to H$\u662f\u4e2a\u540c\u6001, \u5219$G\/Kerf \\cong Imf$.<br \/>\n$\\\\$ \u6709\u4e9b\u6559\u6750\u53c8\u628a\u540c\u6001\u57fa\u672c\u5b9a\u7406\u79f0\u4e3a\u7b2c\u4e00\u540c\u6784\u5b9a\u7406, \u4ece\u540d\u5b57\u4e0a\u6211\u4eec\u4e5f\u53ef\u4ee5\u770b\u51fa\u8fd8\u6709\u7b2c\u4e8c\u540c\u6784\u5b9a\u7406, \u7b2c\u4e09\u540c\u6784\u5b9a\u7406\u7b49\u7b49, \u4e0d\u8fc7\u6211\u4eec\u63a5\u4e0b\u6765\u4e5f\u4e0d\u4f1a\u7528\u5230\u90a3\u4e9b\u4ee3\u6570\u6280\u672f, \u6545\u4e0d\u518d\u8d58\u8ff0. \u540c\u6001\u57fa\u672c\u5b9a\u7406\u5728\u62bd\u8c61\u4ee3\u6570\u4e2d\u975e\u5e38\u91cd\u8981, \u76f4\u63a5\u8bb0\u4f4f\u4e00\u53e5\u8bdd\u5373\u53ef: \u5b9a\u4e49\u57df\u96c6\u5408\u6a21\u53bb\u540c\u6001\u7684\u6838\u4e0e\u540c\u6001\u7684\u50cf\u662f\u540c\u6784\u7684~<\/p>\n<p><strong>4. \u76f8\u5173\u4e60\u9898<\/strong><\/p>\n<p><strong>\u547d\u9898:<\/strong> \u8bbe$f:Z \\to Z$\u662f\u6ee1\u540c\u6001, \u8bc1\u660e$f$\u662f\u540c\u6784.<br \/>\n$\\\\$ <strong>\u8bc1:<\/strong> (Emm\u2026\u2026\u611f\u89c9\u9898\u76ee\u7f3a\u4e86\u6761\u4ef6=.=) \u5373\u4ec5\u9700\u8bc1$f$\u662f\u5355\u540c\u6001\u5373\u53ef, \u7531\u540c\u6001\u5b9a\u4e49\u53ef\u77e5$f(n) = nf(1)$. \u56e0\u6b64\u5f53$f(1) = \\pm 1$\u65f6$f$\u662f\u540c\u6784, \u6b64\u65f6\u547d\u9898\u5f97\u8bc1.<\/p>\n<p>\u63a5\u4e0b\u6765\u53ef\u4ee5\u5f00\u5e72\u57fa\u672c\u7fa4\u5566! \u51b2\u51b2\u51b2!!!<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u4eca\u5929\u7b97\u662f\u56fd\u5e86\u5047\u671f\u7684\u6700\u540e\u4e00\u5929\u4e86, \u867d\u7136\u660e\u5929\u4e5f\u4e0d\u7528\u4e0a\u73ed, \u4f46\u65f6\u95f4\u57fa\u672c\u90fd\u82b1\u5728\u8def\u4e0a\u4e86, \u6240\u4ee5\u6211\u5185\u5fc3\u4e5f\u5c31\u4e0d\u628a\u660e\u5929\u5f53\u5047\u671f\u4e86 &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/10\/09\/group_common_knowledge_review\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u7fa4\u7684\u5e38\u7528\u77e5\u8bc6\u590d\u4e60<\/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\/1206"}],"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=1206"}],"version-history":[{"count":15,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1206\/revisions"}],"predecessor-version":[{"id":2552,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1206\/revisions\/2552"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=1206"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=1206"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=1206"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}