{"id":433,"date":"2023-10-01T18:10:01","date_gmt":"2023-10-01T10:10:01","guid":{"rendered":"https:\/\/qwq.cafe\/?p=433"},"modified":"2023-10-30T10:15:45","modified_gmt":"2023-10-30T02:15:45","slug":"sylow-%e5%ae%9a%e7%90%86","status":"publish","type":"post","link":"https:\/\/qwq.cafe\/?p=433","title":{"rendered":"Sylow \u5b9a\u7406"},"content":{"rendered":"

Sylow \u5b9a\u7406\u662f\u7fa4\u8bba\u4e2d\u7684\u91cd\u8981\u5b9a\u7406, \u4f7f\u6211\u4eec\u66f4\u4e86\u89e3\u6709\u9650\u7fa4\u7684\u7ed3\u6784. \u4f46\u662f\u5176\u8bc1\u660e\u8fc7\u7a0b\u5728\u5f88\u591a\u4e66\u4e0a\u6bd4\u8f83\u7e41\u7410\u800c\u4e14\u8ba9\u4eba\u6478\u4e0d\u7740\u5934\u8111, \u672c\u6587\u6574\u7406\u4e86Advanced Modern Algebra<\/em> \u8fd9\u672c\u4e66\u4e2d\u7684\u8bc1\u660e\u65b9\u6cd5, \u5176\u66f4\u4e3a\u6e05\u6670\u4e14\u81ea\u7136. <\/p>\n

<\/p>\n

\u672c\u6587\u4e2d\u6240\u6709\u7fa4\u5747\u4e3a\u4e58\u6cd5\u7fa4, \u4e14\u5355\u4f4d\u5143\u7528 $1$ \u8868\u793a.<\/p>\n

Sylow \u5b9a\u7406\u7531\u4e09\u6761\u5b9a\u7406\u7ec4\u6210, \u5728\u4e00\u822c\u4e66\u4e0a\u8868\u8fbe\u5982\u4e0b:<\/p>\n

Thm (Sylow \u7b2c\u4e00\u5b9a\u7406).<\/strong> \u8bbe\u7fa4 $G$ \u7684\u9636\u4e3a $p^em$, \u5176\u4e2d $p$ \u4e3a\u7d20\u6570\u4e14 $\\gcd(m,p)=1,e>0$, \u5219\u5bf9\u4e8e $1\\le k\\le e$, $G$ \u4e2d\u5fc5\u6709 $p^k$ \u9636\u5b50\u7fa4. \u5176\u4e2d $p^e$ \u9636\u5b50\u7fa4\u88ab\u79f0\u4e3a $G$ \u7684 Sylow $p$ – \u5b50\u7fa4<\/strong>.<\/p>\n

Thm (Sylow \u7b2c\u4e8c\u5b9a\u7406).<\/strong> \u8bbe\u7fa4 $G$ \u7684\u9636\u4e3a $p^em$, \u5176\u4e2d $p$ \u4e3a\u7d20\u6570\u4e14 $\\gcd(m,p)=1,e>0$, \u5219<\/p>\n

    \n
  1. \u5bf9\u4e8e $1\\le k\\le e$, $G$ \u7684\u4efb\u610f\u4e00\u4e2a $p^k$ \u9636\u5b50\u7fa4\u4e00\u5b9a\u5305\u542b\u5728 $G$ \u7684\u67d0\u4e00\u4e2a Sylow $p$ – \u5b50\u7fa4\u4e2d;<\/li>\n
  2. $G$ \u7684\u4efb\u610f\u4e24\u4e2a Sylow $p$ – \u5b50\u7fa4\u5171\u8f6d.<\/li>\n<\/ol>\n

    Thm (Sylow \u7b2c\u4e09\u5b9a\u7406).<\/strong> \u8bbe\u7fa4 $G$ \u7684\u9636\u4e3a $p^e m$, \u5176\u4e2d $p$ \u4e3a\u7d20\u6570\u4e14 $\\gcd(m,p)=1,e>0$, \u5219 $G$ \u7684 Sylow $p$ – \u5b50\u7fa4\u7684\u4e2a\u6570 $r$ \u6ee1\u8db3 $r\\mid m$ \u4e14
    \n$$r\\equiv 1\\bmod p
    \n$$
    \n\u4f46\u662f Advanced Modern Algebra<\/em> \u4e2d\u7684\u8868\u8ff0\u6709\u4e9b\u8bb8\u4e0d\u540c, \u63a5\u4e0b\u6765\u5c31\u6765\u770b\u770b\u8be5\u4e66\u4e2d\u7684 Sylow \u5b9a\u7406\u53ca\u5176\u8bc1\u660e.<\/p>\n

    \n

    Lemma.<\/strong> $G$ \u662f\u4e00\u4e2a\u6709\u9650 Abel \u7fa4, $p$ \u662f $|G|$ \u7684\u4e00\u4e2a\u7d20\u56e0\u5b50, \u5219 $G$ \u542b\u6709 $p$ \u9636\u5b50\u7fa4.<\/p>\n

    Proof.<\/p>\n

    \u8bbe $|G|=pm$, \u5bf9 $m$ \u8fdb\u884c\u6570\u5b66\u5f52\u7eb3.<\/p>\n

      \n
    • \n

      \u5f53 $m=1$ \u65f6, \u663e\u7136\u6210\u7acb.<\/p>\n<\/li>\n

    • \n

      \u5f53 $m\\ge 2$ \u65f6, \u4efb\u53d6 $G$ \u4e2d\u7684\u4e00\u4e2a\u975e\u5355\u4f4d\u5143\u5143\u7d20 $a$.<\/p>\n

      \u82e5 $|a|=kp$ \u5bf9\u4e8e\u67d0\u4e2a $k\\in \\mathbb{Z}_+$, \u5219 $\\left< a^k\\right>$ \u5373\u4e3a $p$ \u9636\u5b50\u7fa4;<\/p>\n

      \u5426\u5219, \u5373 $p\\nmid |a|$, \u8003\u8651\u5546\u7fa4 $G\/\\left<a\\right>$, \u5176\u9636\u4e3a $pm'$, \u5176\u4e2d $m'=\\frac m{|a|}<m$. <\/p>\n

      \u7531\u5f52\u7eb3\u5047\u8bbe\u77e5 $G\/\\left<a\\right>$ \u4e2d\u5b58\u5728 $p$ \u9636\u5b50\u7fa4 $\\left<[b]\\right>$ ($p$ \u9636\u5b50\u7fa4\u4e00\u5b9a\u4e3a\u5faa\u73af\u7fa4, $b\\in G$, $[b]:=b\\left<a\\right>$).<\/p>\n

      \u8bbe $|b^p|=s,|b|=t$, \u5219\u7531\u4e8e $[b]$ \u4e3a $p$ \u9636\u5143, $\\gcd(s,p)=1$ (\u56e0\u4e3a $b^p\\in\\left< a\\right>$ \u4e14 $p\\nmid |a|$). \u7531\u4e8e $b^{sp}=1$ \u56e0\u6b64 $t\\mid sp$.<\/p>\n

      \u5047\u8bbe $p\\nmid t$, \u5219\u6709 $t\\mid s$, \u5219\u6709 $[b]^s=\\left<a\\right>$ \u4e0e $\\gcd(s,p)=1$ \u4ea7\u751f\u77db\u76fe.<\/p>\n

      \u56e0\u6b64 $p\\mid t$, \u5219 $\\left<b^{t\/p}\\right>$ \u5373\u4e3a $G$ \u7684 $p$ \u9636\u5b50\u7fa4. $\\square$<\/p>\n<\/li>\n<\/ul>\n

      Thm (Gauss).<\/strong> $G$ \u662f\u4e00\u4e2a\u6709\u9650\u7fa4, $p$ \u662f $|G|$ \u7684\u4e00\u4e2a\u7d20\u56e0\u5b50, \u5219 $G$ \u542b\u6709 $p$ \u9636\u5b50\u7fa4.<\/p>\n

      \u8fd9\u91cc\u540c\u6837\u4f7f\u7528\u6570\u5b66\u5f52\u7eb3\u6cd5\u8bc1\u660e.<\/p>\n

      Proof.<\/p>\n

      \u8bbe $|G|=pm$, \u5bf9 $m$ \u8fdb\u884c\u6570\u5b66\u5f52\u7eb3.<\/p>\n

        \n
      • \n

        \u5f53 $m=1$ \u65f6, \u663e\u7136\u6210\u7acb.<\/p>\n<\/li>\n

      • \n

        \u5f53 $m\\ge 2$ \u65f6, \u7528 $Z(G)$ \u8868\u793a $G$ \u7684\u4e2d\u5fc3.<\/p>\n

        \u4efb\u53d6 $a\\in G-Z(G)$, \u8bb0 $\\mathcal{O}_a$ \u8868\u793a $a$ \u7684\u5171\u8f6d\u7c7b, $C(a)$ \u8868\u793a $a$ \u7684\u4e2d\u5fc3\u5316\u5b50\u7fa4.<\/p>\n

        \u82e5 $p\\nmid |\\mathcal O_a|$, \u7531\u8f68\u9053 – \u7a33\u5b9a\u5b50\u5b9a\u7406\u77e5 $|\\mathcal O_a||C(a)|=|G|$. \u56e0\u6b64 $p\\mid |C(a)|$, \u7531\u5f52\u7eb3\u5047\u8bbe\u77e5 $|C(a)|$ \u4e2d\u6709 $G$ \u7684 $p$ \u9636\u5b50\u7fa4.<\/p>\n

        \u82e5 $\\forall a\\in G-Z(G),p\\mid |\\mathcal O_a|$, \u518d\u7ed3\u5408\u5171\u8f6d\u7c7b\u65b9\u7a0b
        \n$$ |G|=|Z(G)|+\\sum_i |\\mathcal O_{a_i}|
        \n$$
        \n\u5f97 $p\\mid|Z(G)|$, \u7531\u4e8e $Z(G)$ \u662f Abel \u7fa4, \u53ef\u7531\u4e4b\u524d\u7684\u5f15\u7406\u77e5\u5176\u4e2d\u6709 $G$ \u7684 $p$ \u9636\u5b50\u7fa4. $\\square$<\/p>\n<\/li>\n<\/ul>\n

        Def.<\/strong> $G$ \u662f\u4e00\u4e2a\u7fa4, $p$ \u662f\u4e00\u4e2a\u7d20\u6570, \u82e5 $G$ \u4e2d\u6240\u6709\u5143\u7d20\u7684\u9636\u90fd\u662f $p$ \u7684\u5e42, \u5219\u79f0 $G$ \u662f\u4e00\u4e2a $p$ – \u7fa4<\/strong>.<\/p>\n

        \u5bb9\u6613\u5f97\u77e5, <\/p>\n

          \n
        • \u4e00\u4e2a\u6709\u9650\u7fa4 $G$ \u662f $p$ – \u7fa4, \u5f53\u4e14\u4ec5\u5f53\u5176\u9636\u662f $p$ \u7684\u5e42.<\/li>\n
        • $p$ \u7fa4\u7684\u5b50\u7fa4\u4e5f\u662f $p$ – \u7fa4.<\/li>\n<\/ul>\n

          Prop.<\/strong> $G$ \u662f\u4e00\u4e2a\u975e\u5e73\u51e1\u6709\u9650 $p$ – \u7fa4, \u5219\u5176\u4e2d\u5fc3 $Z(G)$ \u4e0d\u4e3a $\\{1\\}$ \u4e14\u4e5f\u4e3a\u6709\u9650 $p$ \u7fa4.<\/p>\n

          Proof.<\/p>\n

          \u7531\u5171\u8f6d\u7c7b\u65b9\u7a0b\u77e5
          \n$$|G|=|Z(G)|+\\sum_i[G:C(a_i)]
          \n$$
          \n\u7531\u4e8e $a_i\\notin Z(G)$, $p\\mid[G:C(a_i)]$, \u56e0\u6b64 $p\\mid |Z(G)|$. $\\square$<\/p>\n

          Prop.<\/strong> $G$ \u662f\u4e00\u4e2a\u9636\u4e3a $p^e (e>0)$ \u7684\u7fa4, \u5219 $G$ \u6709 $p^k (0<k\\le e)$ \u9636\u7684\u5b50\u7fa4.<\/p>\n

          Proof.<\/p>\n

          \u5f53 $k=1$ \u65f6\u7531\u9ad8\u65af\u5b9a\u7406\u5f97\u8bc1, \u56e0\u6b64\u53ea\u7528\u8bc1\u660e $k>1$ \u7684\u60c5\u5f62. \u63a5\u4e0b\u6765\u5bf9 $e$ \u8fdb\u884c\u6570\u5b66\u5f52\u7eb3.<\/p>\n

            \n
          • \n

            \u5f53 $e=2$ \u65f6, \u663e\u7136\u6210\u7acb.<\/p>\n<\/li>\n

          • \n

            \u5f53 $e>2$ \u65f6, $Z(G)\\ne\\{1\\}$ \u4e14 $Z(G)$ \u4e3a $p$ – \u7fa4. \u5219\u5b58\u5728 $Z(G)$ \u7684 $p$ \u9636\u5b50\u7fa4 $H$.<\/p>\n

            \u7531\u4e8e $H\\subset Z(G)$, $H$ \u4e3a $G$ \u7684\u6b63\u89c4\u5b50\u7fa4.<\/p>\n

            \u8003\u8651\u5546\u7fa4 $G\/H$, \u7531\u5f52\u7eb3\u5047\u8bbe\u77e5\u5176\u4e2d\u5b58\u5728 $k-1$ \u9636\u7fa4 $K'$.<\/p>\n

            \u7531 The Correspondence Theorem \u77e5\u5b58\u5728 $G$ \u7684\u5b50\u7fa4 $K$ \u4f7f\u5f97 $K\/H=K'$.<\/p>\n

            \u5219 $K$ \u4e3a $G$ \u7684 $p^k$ \u9636\u5b50\u7fa4. $\\square$<\/p>\n<\/li>\n<\/ul>\n

            Def.<\/strong> $G$ \u662f\u4e00\u4e2a\u7fa4, $p$ \u662f\u4e00\u4e2a\u7d20\u6570, $H$ \u662f $G$ \u7684 $p$ – \u5b50\u7fa4. \u82e5\u4e0d\u5b58\u5728 $G$ \u7684\u771f $p$ – \u5b50\u7fa4 $K$ \u4f7f\u5f97 $H\\subsetneqq K$, \u5219\u79f0 $H$ \u662f $G$ \u7684 Sylow $p$ – \u5b50\u7fa4.<\/p>\n

            \u8fd9\u4e48\u5b9a\u4e49 Sylow $p$ – \u5b50\u7fa4\u7684\u597d\u5904\u662f\u5b9a\u4e49\u4e86\u65e0\u9650\u7fa4\u7684\u60c5\u5f62, \u800c\u4e14\u80fd\u4f7f\u5f97\u540e\u9762\u7684\u8bc1\u660e\u53d8\u5f97\u66f4\u7b80\u5355.<\/p>\n

            \u8bc1\u660e Sylow $p$ – \u5b50\u7fa4\u7684\u5b58\u5728\u6027\u53ea\u9700\u8981\u7528 Zorn \u5f15\u7406\u4fbf\u53ef\u4ee5\u5b8c\u6210.<\/p>\n

            Lemma (Zorn).<\/strong> $(X,\\le)$ \u662f\u4e00\u4e2a\u504f\u5e8f\u96c6, \u82e5\u5176\u6bcf\u4e2a\u94fe(\u5168\u5e8f\u5b50\u96c6)\u90fd\u6709\u4e0a\u754c, \u5219 $X$ \u4e2d\u5b58\u5728\u6781\u5927\u5143(\u5373 $\\exists x\\in X,\\nexists c\\in X,x < c$).<\/p>\n

            Zorn \u5f15\u7406\u662f\u96c6\u5408\u8bba\u4e2d\u7684\u91cd\u8981\u5f15\u7406, \u5176\u7b49\u4ef7\u4e8e\u9009\u62e9\u516c\u7406. \u7531\u4e8e\u8fd9\u4e2a\u4e0d\u662f\u672c\u6587\u7684\u91cd\u70b9, \u5728\u6b64\u4e0d\u505a\u8bc1\u660e.<\/p>\n

            Prop.<\/strong> $G$ \u7684 Sylow $p$ – \u5b50\u7fa4\u5b58\u5728.<\/p>\n

            Proof.<\/p>\n

            \u8bbe $M$ \u662f $G$ \u7684\u6240\u6709 $p$ – \u5b50\u7fa4\u6784\u6210\u7684\u96c6\u5408. \u5b9a\u4e49 $M$ \u4e0a\u7684\u5e8f\u5173\u7cfb\u4e3a\u96c6\u5408\u7684\u5305\u542b\u5173\u7cfb.<\/p>\n

            $M$ \u7684\u6bcf\u4e2a\u94fe\u90fd\u6709\u94fe\u5185\u6240\u6709\u5143\u7d20\u4e4b\u5e76\u96c6\u4e3a\u4e0a\u754c, \u7531 Zorn \u5f15\u7406\u77e5, $M$ \u4e2d\u5b58\u5728\u6781\u5927\u5143, \u5373 $G$ \u7684 Sylow $p$ – \u5b50\u7fa4. $\\square$<\/p>\n

            Lemma.<\/strong> $P$ \u662f\u6709\u9650\u7fa4 $G$ \u7684 Sylow $p$ – \u5b50\u7fa4.<\/p>\n

              \n
            1. \u6bcf\u4e2a $P$ \u7684\u5171\u8f6d\u7fa4\u4e5f\u662f $G$ \u7684 Sylow $p$ – \u5b50\u7fa4.<\/li>\n
            2. $[N(P):P]$ \u4e0e $p$ \u4e92\u7d20. ($N(P)$ \u8868\u793a $P$ \u7684\u6b63\u89c4\u5316\u5b50\u7fa4)<\/li>\n
            3. \u82e5 $a\\in G$ \u6ee1\u8db3 $|a|$ \u4e3a $p$ \u7684\u5e42\u4e14 $aPa^{-1}=P$, \u5219\u6709 $a\\in P$.<\/li>\n<\/ol>\n

              Proof.<\/p>\n

                \n
              1. \n

                \u82e5 $P$ \u7684\u5171\u8f6d $aPa^{-1}$ \u4e0d\u662f $G$ \u7684 Sylow $p$ – \u5b50\u7fa4.<\/p>\n

                \u5219\u5b58\u5728 $G$ \u7684 $p$ – \u5b50\u7fa4 $H$ \u4f7f\u5f97 $aPa^{-1}\\subsetneqq H\\Rightarrow P\\subsetneqq a^{-1}Ha$ \u4ea7\u751f\u77db\u76fe.<\/p>\n<\/li>\n

              2. \n

                \u5047\u8bbe $p\\mid [N(P):P]$, \u5219\u7531 Gauss \u5b9a\u7406\u77e5\u5546\u7fa4 $N(P)\/P$ \u5b58\u5728 $p$ \u9636\u5b50\u7fa4 $H'$.<\/p>\n

                \u00a0 \u7531 The Correspondence Theorem \u77e5\u5b58\u5728\u5b50\u7fa4 $H\\subset N(P)$ \u6ee1\u8db3 $H\/P=H'$.<\/p>\n

                \u5219 $H$ \u662f $p$ – \u7fa4\u4e14 $P\\subsetneqq H$ \u4ea7\u751f\u77db\u76fe.<\/p>\n<\/li>\n

              3. \n

                \u56e0\u4e3a $aPa^{-1}=P$, \u6709 $a\\in N(P)$.<\/p>\n

                \u7531 (2) \u53ef\u77e5 $aP=P$, \u56e0\u6b64 $a\\in P$. $\\square$<\/p>\n<\/li>\n<\/ol>\n

                \u81f3\u6b64, \u94fa\u57ab\u5df2\u7ecf\u505a\u8db3, \u63a5\u4e0b\u6765\u5c06\u6b63\u5f0f\u5f00\u59cb\u8bc1\u660e Sylow \u5b9a\u7406.<\/p>\n

                Thm (Sylow \u7b2c\u4e8c\u548c\u7b2c\u4e09\u5b9a\u7406)<\/strong> $G$ \u662f\u4e00\u4e2a\u6709\u9650\u7fa4\u4e14 $|G|=p^{e}m$, \u5176\u4e2d $\\gcd(p,m)=1,e>0$. \u4ee4 $P$ \u4e3a Sylow $p$ – \u5b50\u7fa4.<\/p>\n

                  \n
                1. \n

                  \u6bcf\u4e2a Sylow $p$ – \u5b50\u7fa4\u90fd\u662f $P$ \u7684\u5171\u8f6d\u7fa4.<\/p>\n<\/li>\n

                2. \n

                  \u5982\u679c\u8fd9\u91cc\u6709 $r$ \u4e2a Sylow $p$ – \u5b50\u7fa4, \u5219 $r$ \u662f $|G|\/p^{e}$ \u7684\u56e0\u5b50\u4e14
                  \n$$
                  \nr\\equiv 1\\bmod p
                  \n$$<\/p>\n<\/li>\n<\/ol>\n

                  Proof.<\/p>\n

                  \u4ee4 $M$ \u8868\u793a $P$ \u7684\u6240\u6709\u5171\u8f6d\u7fa4\u6784\u6210\u7684\u96c6\u5408. \u8bbe $M=\\{P_1,P_2,…,P_r\\}$, \u5176\u4e2d $P_1=P,P_i=g_iPg_i^{-1}$ \u4e14 $P_i\\ne P_j(i\\ne j)$.<\/p>\n

                  \u82e5 $Q$ \u662f\u4e00\u4e2a Sylow $p$ – \u5b50\u7fa4, \u8003\u8651 $Q$ \u5728 $M$ \u4e0a\u7684\u4f5c\u7528 $(a,g_iPg_i^{-1})\\mapsto ag_iPg_i^{-1}a^{-1}=ag_iP(g_ia)^{-1}$.<\/p>\n

                    \n
                  • \n

                    \u82e5 $P$ \u7684\u8f68\u9053 $\\mathcal O_P$ \u6ee1\u8db3 $|\\mathcal O_P|=1$, \u5219\u6709 $Q=P$. <\/p>\n

                    \u82e5 $\\forall a\\in P, aP_ia^{-1}=P_i$, \u5219\u7531\u4e4b\u524d\u7684\u5b9a\u7406\u77e5 $a\\in P_i$, \u56e0\u6b64 $P_i=P$.<\/p>\n

                    \u56e0\u6b64\u82e5 $i\\ne 1$, \u5219\u6709 $|\\mathcal O_{P_i}|\\ne 1$<\/p>\n

                    \u518d\u7531\u8f68\u9053 – \u7a33\u5b9a\u5b50\u5b9a\u7406 $|\\mathcal O_{P_i}|=[Q:Q_{P_i}]$ \u5f97\u77e5 $p\\mid|\\mathcal O_{P_i}|$.<\/p>\n

                    \u53c8\u56e0\u4e3a
                    \n$$ r=|M|=|\\mathcal O_P|+\\sum_{i\\ne 1}|\\mathcal O_{P_i}|=1+\\sum_{i\\ne 1}|\\mathcal O_{P_i}|
                    \n$$
                    \n\u5bf9 $p$ \u53d6\u4f59\u5f97
                    \n$$
                    \nr\\equiv 1\\bmod p
                    \n$$<\/p>\n<\/li>\n

                  • \n

                    \u82e5 $P$ \u7684\u8f68\u9053 $\\mathcal O_P$ \u4e0d\u6ee1\u8db3 $|\\mathcal O_P|=1$. \u5047\u8bbe $Q\\notin M$.<\/p>\n

                    \u5219\u4e0e\u4e0a\u9762\u540c\u7406\u53ef\u77e5\u6bcf\u4e2a $P_i$ \u7684\u8f68\u9053\u90fd\u6709 $p\\mid|\\mathcal O_{P_i}|$.<\/p>\n

                    \u5219
                    \n$$ r=|M|=|\\mathcal O_P|+\\sum_{i\\ne 1}|\\mathcal O_{P_i}|
                    \n$$
                    \n\u5bf9 $p$ \u53d6\u4f59\u5f97
                    \n$$
                    \nr\\equiv 0\\bmod p
                    \n$$
                    \n\u4ea7\u751f\u77db\u76fe. \u56e0\u6b64 $Q\\in M$.<\/p>\n<\/li>\n<\/ul>\n

                    \u7efc\u4e0a\u6240\u8ff0, $M$ \u5373\u6240\u6709 Sylow $p$ – \u5b50\u7fa4\u7684\u96c6\u5408. \u63a5\u4e0b\u6765\u8fd8\u9700\u8bc1\u660e $r$ \u662f $|G|\/p^{e}$ \u7684\u56e0\u5b50.<\/p>\n

                    \u7531\u4e8e\u6240\u6709 Sylow $p$ – \u5b50\u7fa4\u90fd\u4e0e $P$ \u5171\u8f6d, \u56e0\u6b64 $r=|M|=[G:N(P)]$. \u5219\u6709 $r\\mid |G|$.<\/p>\n

                    \u800c $r$ \u53c8\u4e0e $p$ \u4e92\u7d20(\u56e0\u4e3a\u5bf9\u53d6\u4f59\u4e3a 1), \u56e0\u6b64 $r$ \u662f $|G|\/p^{e}$ \u7684\u56e0\u5b50.<\/p>\n

                    \u56e0\u6b64 (2) \u6210\u7acb. $\\square$<\/p>\n

                    Thm (Sylow \u7b2c\u4e00\u5b9a\u7406).<\/strong> \u8bbe\u7fa4 $G$ \u7684\u9636\u4e3a $p^em$, \u5176\u4e2d $p$ \u4e3a\u7d20\u6570\u4e14 $\\gcd(m,p)=1,e>0$, \u5219 $G$ \u7684 Sylow $p$ – \u5b50\u7fa4\u4e3a $p^e$ \u9636.<\/p>\n

                    Proof.<\/p>\n

                    \u8bbe $H$ \u4e3a $G$ \u7684 Sylow $p$ – \u5b50\u7fa4, \u5219\u6709 $[G:H]=[G:N(H)][N(H):H]$.<\/p>\n

                    \u7531\u4e4b\u524d\u7684\u5f15\u7406\u53ef\u77e5 $p\\nmid[N(H):H]$.<\/p>\n

                    \u53c8\u7531 Sylow \u7b2c\u4e8c\u548c\u7b2c\u4e09\u5b9a\u7406\u53ef\u77e5 $[N(H):H]$ \u4e0e $p$ \u4e92\u7d20.<\/p>\n

                    \u56e0\u6b64 $[G:H]$ \u4e0e $p$ \u4e92\u7d20, \u800c $H$ \u7684\u9636\u53c8\u4e3a $p$ \u7684\u5e42, \u56e0\u6b64 $|H|=p^e$. $\\square$<\/p>\n","protected":false},"excerpt":{"rendered":"

                    Sylow \u5b9a\u7406\u662f\u7fa4\u8bba\u4e2d\u7684\u91cd\u8981\u5b9a\u7406, \u4f7f\u6211\u4eec\u66f4\u4e86\u89e3\u6709\u9650\u7fa4\u7684\u7ed3\u6784. \u4f46\u662f\u5176\u8bc1\u660e\u8fc7\u7a0b\u5728\u5f88\u591a\u4e66\u4e0a\u6bd4\u8f83\u7e41\u7410\u800c\u4e14\u8ba9\u4eba\u6478\u4e0d […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[11,8],"class_list":["post-433","post","type-post","status-publish","format-standard","hentry","category-6","tag-11","tag-8"],"_links":{"self":[{"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/posts\/433","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/qwq.cafe\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=433"}],"version-history":[{"count":16,"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/posts\/433\/revisions"}],"predecessor-version":[{"id":452,"href":"https:\/\/qwq.cafe\/index.php?rest_route=\/wp\/v2\/posts\/433\/revisions\/452"}],"wp:attachment":[{"href":"https:\/\/qwq.cafe\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=433"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/qwq.cafe\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=433"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/qwq.cafe\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=433"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}