\n($\\Leftarrow$)<\/p>\n
\u5de6\u5355\u4f4d\u5143\u548c\u53f3\u5355\u4f4d\u5143\u76f8\u7b49: $ee=e\\Rightarrow e(xx^{-1})=xx^{-1}\\Rightarrow exx^{-1}(x^{-1})^{-1}=xx^{-1}(x^{-1})^{-1}\\Rightarrow ex=x$.<\/p>\n
\u5de6\u9006\u5143\u548c\u53f3\u9006\u5143\u76f8\u7b49: $x^{-1}xx^{-1}=ex^{-1}=x^{-1}\\Rightarrow x^{-1}xx^{-1}(x^{-1})^{-1}=x^{-1}(x^{-1})^{-1}=e\\Rightarrow x^{-1}x=e$. $\\square$<\/p>\n<\/li>\n<\/ul>\n
\u5b9a\u4e491.1.3 \u7fa4\u7684\u7b49\u4ef7\u5b9a\u4e492<\/h4>\n
\u8bbe $G$ \u662f\u4e00\u4e2a\u975e\u7a7a\u96c6\u5408, \u5982\u679c\u5728 $G$ \u4e0a\u5b9a\u4e49\u4e86\u4e00\u4e2a\u4ee3\u6570\u8fd0\u7b97, \u4e14 $\\forall a,b\\in G$, \u65b9\u7a0b $ax=b$ \u548c $ya=b$ \u5747\u6709\u89e3.<\/p>\n
\u7b49\u4ef7\u8bc1\u660e ($1.1.2\\Leftrightarrow 1.1.3$): <\/strong><\/p>\nProof.<\/p>\n
\n- \n
($\\Rightarrow$)<\/p>\n
$\\forall a,b\\in G$, \u4ee4 $x=a^{-1}b,y=ba^{-1}$, \u5219\u6709 $ax=b,ya=b$. \u5373\u65b9\u7a0b $ax=b$ \u548c $ya=b$ \u5747\u6709\u89e3.<\/p>\n<\/li>\n
- \n
($\\Leftarrow$)<\/p>\n
\u4efb\u53d6 $a\\in G$, \u4ee4 $e$ \u4e3a $ax=a$ \u7684\u89e3.<\/p>\n
\n- \n
\u5148\u8bc1\u660e $\\forall b\\in G,be=b$: <\/p>\n
\u4ee4 $x_0$ \u4e3a $xa=b$ \u7684\u89e3, \u5219\u6709 $be=(x_0 a)e=x_0(ae)=x_0a=b$.<\/p>\n<\/li>\n
- \n
\u518d\u8bc1\u660e\u53f3\u9006\u5143\u5b58\u5728: <\/p>\n
$\\forall b\\in G$, \u65b9\u7a0b $bx=e$ \u6709\u89e3\u6240\u4ee5\u53f3\u9006\u5143\u5b58\u5728. $\\square$<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n
\u5b9a\u4e49 1.1.4 \u534a\u7fa4 (Semigroup)\u3001\u5e7a\u534a\u7fa4 (Monoid)<\/h4>\n
\u8bbe $G$ \u662f\u4e00\u4e2a\u975e\u7a7a\u96c6\u5408, \u5982\u679c\u5728 $G$ \u4e0a\u5b9a\u4e49\u4e86\u4e00\u4e2a\u4ee3\u6570\u8fd0\u7b97.<\/p>\n
\n- \n
\u7ed3\u5408\u5f8b: $\\forall a,b,c\\in G,\\ (ab)c=a(bc)$.<\/p>\n<\/li>\n
- \n
\u5b58\u5728\u5355\u4f4d\u5143: $\\exists e\\in G,\\forall a\\in G,ae=ea=a$.<\/p>\n<\/li>\n<\/ol>\n
\u5982\u679c\u8fd9\u4e2a\u4ee3\u6570\u8fd0\u7b97\u6ee1\u8db3 (1), \u5219\u79f0 $G$ \u4e3a\u534a\u7fa4<\/strong>, \u5982\u679c\u6ee1\u8db3 (1) \u548c (2), \u5219\u79f0 $G$ \u662f\u5e7a\u534a\u7fa4<\/strong>.<\/p>\n\u5b9a\u4e491.1.5 \u6709\u9650\u7fa4\u3001\u65e0\u9650\u7fa4<\/h4>\n
\u5982\u679c\u7fa4 $G$ \u662f\u6709\u9650\u96c6\u5408, \u5219\u79f0 $G$ \u662f\u6709\u9650\u7fa4, \u5b83\u7684\u5143\u7d20\u4e2a\u6570\u79f0\u4e3a $G$ \u7684\u9636 (order), \u8bb0\u4f5c $|G|$.<\/p>\n
\u5982\u679c\u7fa4 $G$ \u662f\u65e0\u9650\u96c6\u5408, \u5219\u79f0 $G$ \u662f\u65e0\u9650\u7fa4.<\/p>\n
\u5b9a\u4e491.1.6 \u5b50\u7fa4 (Subgroup)<\/h4>\n
$G$ \u662f\u4e00\u4e2a\u7fa4, \u5982\u679c $H\\subset G\\land H\\ne \\emptyset$, \u4e14 $H$ \u5173\u4e8e\u7fa4 $G$ \u7684\u4ee3\u6570\u8fd0\u7b97\u6784\u6210\u4e00\u4e2a\u7fa4 (\u8be5\u8fd0\u7b97\u9700\u8981\u5728 $H$ \u4e0a\u5c01\u95ed, \u5373 $\\forall a,b\\in H,ab\\in H$), \u5219\u79f0 $H$ \u662f $G$ \u7684\u5b50\u7fa4, \u8bb0\u4f5c $H<G$.<\/p>\n
\u547d\u98981.1.1 \u4e00\u4e2a\u7fa4\u7684\u5b50\u7fa4\u548c\u5176\u5177\u6709\u76f8\u540c\u7684\u5355\u4f4d\u5143\u4e14\u540c\u4e00\u5143\u7d20\u5728\u4e24\u7fa4\u4e2d\u7684\u9006\u5143\u76f8\u540c<\/h4>\n
Proof.<\/p>\n
\u8bbe $G$ \u662f\u4e00\u4e2a\u7fa4, $H<G$.<\/p>\n
\n- \n
\u5177\u6709\u76f8\u540c\u7684\u5355\u4f4d\u5143:<\/p>\n
\u4efb\u53d6 $a\\in H$, \u5219 $ae_{H}=a\\Rightarrow a^{-1}ae_H=a^{-1}a\\Rightarrow ee_H=e\\Rightarrow e_H=e$.<\/p>\n<\/li>\n
- \n
\u540c\u4e00\u5143\u7d20\u5728\u4e24\u7fa4\u4e2d\u7684\u9006\u5143\u76f8\u540c<\/p>\n
\u4efb\u53d6 $a\\in H$, \u5219 $aa^{-1}_H=e_H=e\\Rightarrow a^{-1}_H=a^{-1}$. $\\square$<\/p>\n<\/li>\n<\/ul>\n
\u547d\u98981.1.2 $G$ \u662f\u4e00\u4e2a\u7fa4, $H\\subset G$ \u4e14 $H\\ne \\emptyset$, \u5219 $H<G\\Leftrightarrow (\\forall a,b\\in H, ab^{-1}\\in H)$<\/h4>\n
Proof.<\/p>\n
\n- \n
($\\Rightarrow$)<\/p>\n
\u4efb\u53d6 $a,b\\in H$, \u7531\u547d\u9898 1.1.1 \u77e5 $b^{-1}\\in \tH$, \u6240\u4ee5 $ab^{-1}\\in H$.<\/p>\n<\/li>\n
- \n
($\\Leftarrow$)<\/p>\n
\u4efb\u53d6 $a\\in H$, \u6709 $e=aa^{-1}\\in H$.<\/p>\n
\u518d\u4efb\u53d6 $b\\in H$, \u6709 $b^{-1}=eb^{-1}\\in H$.<\/p>\n
\u6240\u4ee5 $H$ \u5173\u4e8e $\\cdot$ \u6784\u6210\u7fa4, $H< G$. $\\square$<\/p>\n<\/li>\n<\/ul>\n
\u547d\u98981.1.3 \u8bbe $H,K$ \u90fd\u662f $G$ \u7684\u5b50\u7fa4, \u5219 $H\\cap K$ \u662f $G$ \u7684\u5b50\u7fa4.<\/h4>\n
Proof.<\/p>\n
\u4efb\u53d6 $a,b\\in H\\cap K$, \u5219\u6709 $ab^{-1}\\in H$ \u4e14 $ab^{-1}\\in K$<\/p>\n
\u6240\u4ee5 $ab^{-1}\\in H\\cap K$, \u7531\u547d\u9898 1.1.2 \u77e5 $H\\cap K$ \u662f $G$ \u7684\u5b50\u7fa4. $\\square$<\/p>\n
\u547d\u98981.1.4 \u8bbe $H,K$ \u90fd\u662f\u7fa4 $G$ \u7684\u5b50\u7fa4, \u4ee4 $HK:=\\{hk|h\\in H,k\\in K\\}$, \u5219 $HK$ \u4e3a\u5b50\u7fa4\u5f53\u4e14\u4ec5\u5f53 $HK=KH$.<\/h4>\n
Proof.<\/p>\n
\n- \n
($\\Rightarrow$)<\/p>\n
\u4efb\u53d6 $h\\in H,k\\in K$, \u5219<\/p>\n
\u56e0\u4e3a $h^{-1}k^{-1}\\in HK\\land HK$ \u662f\u7fa4 $\\Rightarrow (h^{-1}k^{-1})^{-1}\\in HK\\Rightarrow kh\\in HK$.<\/p>\n
\u6240\u4ee5 $KH\\subset HK$.<\/p>\n
\u56e0\u4e3a $k^{-1}h^{-1}\\in HK\\Rightarrow \\exists h_1\\in H,k_1\\in K, h_1k_1=k^{-1}h^{-1}\\Rightarrow hk=(h_1k_1)^{-1}=k_1^{-1}h_1^{-1}\\in KH$<\/p>\n
\u6240\u4ee5 $HK\\subset KH$.<\/p>\n
\u6240\u4ee5 $HK=KH$.<\/p>\n<\/li>\n
- \n
($\\Leftarrow$)<\/p>\n
\u4efb\u53d6 $h_1,h_2\\in H,k_1,k_2\\in K$<\/p>\n
\u7531 $HK=KH$ \u77e5 $k_2^{-1}h_2^{-1}\\in HK$ \u6240\u4ee5 $\\exists h_3\\in H,k_3\\in K,h_3k_3=k_2^{-1}h_2^{-1}$.<\/p>\n
\u6240\u4ee5 $h_1k_1k_2^{-1}h_2^{-1}=h_1k_1h_3k_3$. \u7531 $HK=KH$ \u77e5 $\\exists h_4\\in H,k_4\\in K,h_4k_4=k_1h_3$<\/p>\n
\u6240\u4ee5 $h_1k_1k_2^{-1}h_2^{-1}=h_1h_4k_4k_3\\in HK$. \u7531\u547d\u9898 1.1.2 \u77e5 $HK$ \u4e3a $G$ \u7684\u5b50\u7fa4. $\\square$<\/p>\n<\/li>\n<\/ul>\n
\u5b9a\u4e491.1.7 \u5faa\u73af\u7fa4 (Cyclic Group)<\/h4>\n
$G$ \u662f\u4e00\u4e2a\u7fa4, \u5982\u679c $\\exists \\xi\\in G,\\forall a\\in G,\\exists k\\in \\mathbb{Z},\\xi^k=a$. \u5219\u79f0 $G$ \u662f\u4e00\u4e2a\u5faa\u73af\u7fa4, \u8bb0\u4f5c $\\left< \\xi\\right> $. \u5176\u4e2d, $\\xi$ \u88ab\u79f0\u4e3a $G$ \u7684\u751f\u6210\u5143<\/strong> (generator).<\/p>\n\u5b9a\u4e491.1.8 \u5143\u7d20\u7684\u9636 (Order)<\/h4>\n
\u4e00\u4e2a\u5143\u7d20\u7684\u9636\u5373\u80fd\u4f7f\u5f97 $a^n=e$ \u7684\u6700\u5c0f\u6b63\u6574\u6570 $n$, \u8bb0\u4f5c $|a|=n$.<\/p>\n
\u5982\u679c $\\forall n\\in \\mathbb Z^+,a^n\\ne e$, \u5219\u8bb0\u4f5c $|a|=\\infty$.<\/p>\n
\u5982\u679c $|a|\\ne \\infty$, \u663e\u7136\u6709 $\\left|\\left<a\\right>\\right|=|a|$.<\/p>\n
\u5982\u679c $a^n=e$, \u5219\u6709 $|a|\\mid n$.<\/p>\n
Proof.<\/p>\n
\u82e5 $a^n=e$, \u505a\u5e26\u4f59\u9664\u6cd5\u5f97 $n=k|a|+r$.<\/p>\n
\u5219\u6709 $e=a^n=a^{k|a|+r}=a^r$.<\/p>\n
\u7531\u4e8e $r<|a|$ \u4e14 $a^r=e$, \u6240\u4ee5 $r=0$. $\\square$<\/p>\n
\u547d\u98981.1.5 $G$ \u662f\u4e00\u4e2a\u7fa4, $a,b\\in G$, \u82e5 $(|a|,|b|)=1$ \u4e14 $ab=ba$, \u5219 $|ab|=|a||b|$<\/h4>\n
Proof.<\/p>\n
\u8bbe $n=|a|,m=|b|,s=|ab|$.<\/p>\n
\u505a\u5e26\u4f59\u9664\u6cd5\u5f97 $s=k_1n+r_1,s=k_2m+r_2$.<\/p>\n
$e=(ab)^s=a^sb^s=a^{k_1n+r_1}b^{k_2m+r_2}=a^{r_1}b^{r_2}$.<\/p>\n
\u5219\u6709 $a^{r_1}=b^{-r_2}\\Rightarrow (a^{r_1})^m=(b^{-r_2})^m\\Rightarrow a^{r_1m}=b^{-r_2m}=e$.<\/p>\n
\u6240\u4ee5 $n\\mid r_1m$ \u53c8\u56e0\u4e3a $(n,m)=1$, \u6240\u4ee5 $n\\mid r_1$.<\/p>\n
\u53c8\u7531\u4e8e $r_1<n$ \u6240\u4ee5 $r_1=0$.<\/p>\n
\u540c\u7406\u5f97 $r_2=0$.<\/p>\n
\u6240\u4ee5 $m\\mid s$ \u4e14 $n\\mid s$ $\\Rightarrow mn\\mid s\\Rightarrow s\\ge mn$.<\/p>\n
\u53c8\u56e0\u4e3a $(ab)^{mn}=a^{mn}b^{mn}=e$, \u6240\u4ee5 $s=mn$. $\\square$<\/p>\n
\u547d\u98981.1.6 \u82e5 $G$ \u662f\u4e00\u4e2a\u6709\u9650 Abel \u7fa4, \u5219 $\\exists x\\in G,\\forall a\\in G,|a|\\mid |x|$<\/h4>\n
Proof.<\/p>\n
\u4ee4 $x$ \u4e3a $G$ \u4e2d\u9636\u6700\u5927\u7684\u5143\u7d20, \u4e14 $|x|=n$.<\/p>\n
\u5047\u8bbe $\\exists a\\in G$, $|a|=m$ \u4e14 $m\\nmid n$.<\/p>\n
\u5373 $\\exists p$ ($p$ \u662f\u7d20\u6570), $n=k_1p^{r_1},m=k_2p^{r_2}(k_1,k_2,r_1,r_2\\in \\mathbb N,p\\nmid k_1,p\\nmid k_2)$, $r_2>r_1$.<\/p>\n
\u7531\u4e8e $|a|=k_1p^{r_1}$ \u6240\u4ee5 $|a^{p^{r_1}}|=k_1$.<\/p>\n
\u7531\u4e8e $|b|=k_2p^{r_2}$ \u6240\u4ee5 $|b^{k_2}|=p^{r_2}$.<\/p>\n
\u56e0\u4e3a $p\\nmid k_1\\Rightarrow (k_1,p^{r_2})=1$.<\/p>\n
\u7531\u547d\u9898 1.1.2 \u77e5 $|a^{r_1}b^{k_2}|=k_1p^{r_2}>k_1p^{r_1}=n$, \u4e0e $x$ \u662f\u9636\u6700\u5927\u7684\u5143\u7d20\u4ea7\u751f\u77db\u76fe.<\/p>\n
\u6240\u4ee5 $\\forall a\\in G,|a|\\mid n$. $\\square$<\/p>\n
\u547d\u98981.1.7 \u82e5 $G$ \u4e3a\u6709\u9650 Abel \u7fa4, \u5219 $G$ \u4e3a\u5faa\u73af\u7fa4\u5f53\u4e14\u4ec5\u5f53 $\\forall m\\in\\mathbb {Z}^+$, \u65b9\u7a0b $x^m$ \u7684\u6839\u7684\u4e2a\u6570\u4e0d\u8d85\u8fc7 $m$.<\/h4>\n
Proof.<\/p>\n
\n- \n
($\\Rightarrow$)<\/p>\n
\u56e0\u4e3a $G$ \u662f\u5faa\u73af\u7fa4, \u6240\u4ee5 $\\exists a\\in G$, \u4f7f\u5f97 $G=\\left<a\\right>$.<\/p>\n
\u8bbe $n=|a|$.<\/p>\n
\u5219\u65b9\u7a0b $x^m=e$ \u7684\u89e3\u7684\u4e2a\u6570\u5373\u5173\u4e8e $k$ \u7684\u65b9\u7a0b $a^{km}=e\\ (0\\le k<n)$ \u7684\u89e3\u7684\u4e2a\u6570.<\/p>\n
\u5219\u6709 $n|km$, \u5373 $\\frac{n}{(m,n)}|k$, \u5728\u8303\u56f4\u5185\u6ee1\u8db3\u6761\u4ef6\u7684\u6570\u6709 $n\\cdot \\frac{(m,n)}{n}=(m,n)$ \u4e2a, \u800c $(m,n)\\le m$.<\/p>\n<\/li>\n
- \n
($\\Leftarrow$)<\/p>\n
\u7531\u547d\u9898 1.1.6 \u53ef\u77e5, $\\exists a\\in G$, $|a|$ \u7684\u9636\u662f\u5176\u5b83\u6240\u6709\u5143\u7d20\u7684\u9636\u7684\u500d\u6570.<\/p>\n
\u8bbe $n=|a|$. \u7531\u4e8e $n$ \u662f\u5176\u5b83\u6240\u6709\u5143\u7d20\u7684\u9636\u7684\u500d\u6570, \u6240\u4ee5 $\\forall x\\in G,x^n=e$.<\/p>\n
\u6240\u4ee5 $\\forall x\\in G,x$ \u90fd\u662f\u65b9\u7a0b $x^n=e$ \u7684\u89e3, \u5176\u4e2a\u6570\u4e0d\u8d85\u8fc7 $n$ \u4e2a\u6240\u4ee5 $|G|\\le n$.<\/p>\n
\u53c8\u56e0\u4e3a $\\left|\\left< a\\right>\\right|=|a|=n$ \u4e14 $\\left<a\\right>\\subset G$, \u6240\u4ee5 $|G|=\\left|\\left<a\\right>\\right|=n$, \u6240\u4ee5 $G=\\left<a\\right>$. $\\square$<\/p>\n<\/li>\n<\/ul>\n
\u5b9a\u4e49 1.1.9 \u751f\u6210\u5b50\u7fa4<\/h4>\n
\u8bbe $G$ \u662f\u4e00\u4e2a\u7fa4, $S$ \u662f $G$ \u7684\u975e\u7a7a\u5b50\u96c6. \u8bb0
\n$$\\left< S\\right>=\\{a_1^{\\varepsilon_1}a_2^{\\varepsilon_2}\\cdots a_s^{\\varepsilon_s}\\in S|a_1,a_2,\\dots,s_s\\in S,\\varepsilon_1,\\varepsilon_2,\\dots,\\varepsilon_n=\\pm 1,s\\in \\mathbb{Z}^+\\}
\n$$
\n\u663e\u7136 $\\left<S\\right>$ \u662f $G$ \u7684\u4e00\u4e2a\u5b50\u7fa4, \u79f0\u4e3a $G$ \u7684\u7531 $S$ \u751f\u6210\u7684\u5b50\u7fa4 (subgroup generated by $S$). <\/p>\n
\u7279\u522b\u5730, \u5982\u679c $G=\\left<S\\right>$, \u5219\u79f0\u7fa4 $G$ \u662f\u7531 $S$ \u751f\u6210\u7684, $S$ \u53eb\u505a $G$ \u7684\u4e00\u4e2a\u751f\u6210\u5143\u96c6 (generator set), \u5982\u679c $S=\\{a_1,a_2,\\dots,a_n\\}$ \u662f\u4e00\u4e2a\u6709\u9650\u96c6, \u90a3\u4e48\u5c31\u79f0 $G$ \u662f\u6709\u9650\u751f\u6210\u7684 (finitely generated), \u8bb0\u4f5c $G=\\left<a_1,a_2,\\dots,a_n\\right>$.<\/p>\n
\u4f8b:<\/p>\n
\n- \u5faa\u73af\u7fa4\u662f\u7531\u4e00\u4e2a\u5143\u7d20\u751f\u6210\u7684\u7fa4.<\/li>\n
- \u4e8c\u9762\u4f53\u7fa4 (dihedral group) $D_n:=\\left<\\rho_1,\\tau_0\\right>$, \u5176\u4e2d $|\\rho_1|=n,|\\tau_0|=2,\\rho_1\\tau_0\\rho_1=\\tau_0$.<\/li>\n<\/ul>\n
1.2 \u7f6e\u6362\u3001\u5bf9\u79f0\u7fa4<\/h3>\n\u5b9a\u4e491.2.1 \u7f6e\u6362 (Permutation)<\/h4>\n
\u975e\u7a7a\u96c6\u5408 $\\Omega$ \u5230\u81ea\u8eab\u7684\u6240\u6709\u53cc\u5c04\u7ec4\u6210\u7684\u96c6\u5408, \u5bf9\u4e8e\u6620\u5c04\u7684\u4e58\u6cd5\u6210\u4e00\u4e2a\u7fa4, \u79f0\u5b83\u4e3a\u96c6\u5408 $\\Omega$ \u7684\u5168\u53d8\u6362\u7fa4<\/strong> (full transformation group), \u8bb0\u4f5c $S_{\\Omega}$.<\/p>\n\u7279\u522b\u7684, \u5f53 $\\Omega$ \u662f\u6709\u9650\u96c6\u5408\u65f6, $\\Omega$ \u5230\u81ea\u8eab\u7684\u4e00\u4e2a\u53cc\u5c04\u53eb\u505a $\\Omega$ \u7684\u4e00\u4e2a\u7f6e\u6362<\/strong>. \u8bbe $\\Omega$ \u542b\u6709 $n$ \u4e2a\u5143\u7d20, \u4e0d\u59a8\u8bbe $\\Omega=\\{1,2,\\dots,n\\}$, \u8fd9\u65f6 $\\Omega$ \u7684\u4e00\u4e2a\u7f6e\u6362\u79f0\u4e3a $n$ \u5143\u7f6e\u6362<\/strong> (permutation on n letters), \u5e76\u4e14\u79f0 $\\Omega$ \u7684\u5168\u53d8\u6362\u7fa4\u4e3a $n$ \u5143\u5bf9\u79f0\u7fa4<\/strong> (symmetric group on n letters), \u8bb0\u4f5c $S_n$.<\/p>\n1.2 \u8282\u672a\u5b8c\u5f85\u7eed<\/em>
\n\u624d\u4e0d\u662f\u61d2\u5f97\u5199\u5462!<\/del><\/p>\n1.3 \u966a\u96c6\u3001\u5546\u96c6<\/h3>\n\u5b9a\u4e491.3.1 \u966a\u96c6 (Coset)<\/h4>\n
$G$ \u662f\u4e00\u4e2a\u7fa4, $H<G$, \u4efb\u53d6 $a\\in G$, \u79f0 $aH:=\\{ah|h\\in H\\}$ \u4e3a $H$ \u7684\u4e00\u4e2a\u5de6\u966a\u96c6<\/strong> (left coset), \u79f0 $Ha:=\\{ha|h\\in H\\}$ \u4e3a $H$ \u7684\u4e00\u4e2a\u53f3\u966a\u96c6<\/strong> (right coset).<\/p>\n$(ah_1=ah_2\\Rightarrow h_1=h_2)\\Rightarrow |aH|=|H|$.<\/p>\n
\u547d\u98981.3.1\u53ca1.3.2\u7701\u7565 $G$\u662f\u4e00\u4e2a\u7fa4, $H<G$\u7684\u524d\u63d0\u6761\u4ef6.<\/strong><\/p>\n\u547d\u98981.3.1 $aH=bH\\Leftrightarrow b^{-1}a\\in H$<\/h4>\n
Proof.<\/p>\n
\n- \n
($\\Rightarrow $)<\/p>\n
\u4efb\u53d6 $ah_1\\in aH$, \u5219\u6709 $ah_1\\in bH$, \u5219\u53ef\u4ee5\u5f97\u5230 $\\exists h_2\\in H,ah_1=bh_2$.<\/p>\n
\u90a3\u4e48 $b^{-1}a=h_2h_1^{-1}\\in H$.<\/p>\n<\/li>\n
- \n
($\\Leftarrow$)<\/p>\n
\u4efb\u53d6 $ah\\in aH$, \u6709 $ah=a(a^{-1}b)(b^{-1}a)h=b(b^{-1}a)h$.<\/p>\n
\u56e0\u4e3a $(b^{-1}a)h\\in H$ \u6240\u4ee5 $ah=b(b^{-1}a)h\\in bH$.<\/p>\n
\u6240\u4ee5 $aH\\subset bH$, \u540c\u7406\u5f97 $bH\\subset aH$, \u6240\u4ee5\u6709 $aH=bH$. $\\square$<\/p>\n<\/li>\n<\/ul>\n
\u547d\u98981.3.2 $aH= bH\\Leftrightarrow aH\\cap bH\\ne \\emptyset$<\/h4>\n
(\u8be5\u547d\u9898\u7b49\u4ef7\u4e8e $aH\\ne bH\\Leftrightarrow aH\\cap bH= \\emptyset$)<\/p>\n
Proof.<\/p>\n
\n- \n
($\\Rightarrow $)<\/p>\n
\u663e\u7136.<\/p>\n<\/li>\n
- \n
($\\Leftarrow$)<\/p>\n
\u56e0\u4e3a $aH\\cap bH\\ne \\emptyset$, \u4efb\u53d6 $c\\in aH\\cap bH$.<\/p>\n
\u5219 $\\exists h_1,h_2\\in H,ah_1=bh_2=c$.<\/p>\n
\u90a3\u4e48 $ab^{-1}=h_2h_1^{-1}\\in H$, \u7531\u547d\u9898 1.3.1 \u77e5, $aH=bH$. $\\square$<\/p>\n<\/li>\n<\/ul>\n
\u5b9a\u4e491.3.2 \u5546\u96c6 (Quotient Set)<\/h4>\n
$(G\/H)_l:=\\{aH|a\\in G\\}$ \u79f0\u4e3a $G$ \u5173\u4e8e $H$ \u7684\u5de6\u5546\u96c6<\/strong> (left quotient set).<\/p>\n$(G\/H)_r:=\\{Ha|a\\in G\\}$ \u79f0\u4e3a $G$ \u5173\u4e8e $H$ \u7684\u53f3\u5546\u96c6<\/strong> (right quotient set).<\/p>\n\u7531\u547d\u9898 1.3.2 \u77e5 $G=\\coprod_{A\\in(G\/H)_l} A$. ( $\\coprod$ \u8868\u793a\u4e0d\u4ea4\u5e76)<\/p>\n
\u5b9a\u4e49 $[G:H]:=|(G\/H)_l|$.<\/p>\n
\u547d\u98981.3.3 (Lagrange \u5b9a\u7406) $G$ \u662f\u4e00\u4e2a\u6709\u9650\u7fa4, $H<G$, \u5219$|G|=|H|[G:H]$.<\/h4>\n
Lagrange \u5b9a\u7406\u7684\u53e6\u4e00\u79cd\u8868\u793a\u65b9\u5f0f\u662f $|H|\\mid|G|$.<\/p>\n
Proof.<\/p>\n
\u56e0\u4e3a $G=\\coprod_{A\\in(G\/H)_l} A$, \u6240\u4ee5 $|G|=\\sum_{A\\in(G\/H)_l} |A|=\\sum_{A\\in(G\/H)_l} |H|=|H|[G:H]$. $\\square$<\/p>\n
\u547d\u98981.3.4 \u5982\u679c $G$ \u662f\u4e00\u4e2a\u6709\u9650\u7fa4, \u5219\u6709 $\\forall a\\in G,a^{|G|}=e$.<\/h4>\n
Proof.<\/p>\n
$\\left<a\\right>$ \u662f $G$ \u7684\u5b50\u7fa4, \u6240\u4ee5\u7531 Lagrange \u5b9a\u7406\u77e5 $\\left|\\left<a\\right>\\right|\\mid|G|$.<\/p>\n
\u53c8\u56e0\u4e3a $|a|=\\left|\\left<a\\right>\\right|$, \u6240\u4ee5 $|a|\\mid |G|$.<\/p>\n
\u6240\u4ee5 $a^{|G|}=e$. $\\square$<\/p>\n
\u547d\u98981.3.5 (Fermat \u5c0f\u5b9a\u7406) \u5982\u679c $p$ \u662f\u7d20\u6570, \u4e14 $p\\not\\mid a$, \u5219\u6709 $a^{p-1}\\equiv 1\\pmod p$.<\/h4>\n
Proof.<\/p>\n
\u56e0\u4e3a $p$ \u662f\u7d20\u6570, \u4e14 $p\\not\\mid a$, \u6240\u4ee5 $[a]\\in \\mathbb {Z}_p^*$.<\/p>\n
\u7531\u4e8e $\\mathbb {Z}_p^*$ \u5173\u4e8e\u5269\u4f59\u7c7b\u4e58\u6cd5\u6784\u6210\u7fa4, \u6240\u4ee5 $[a]^{|\\mathbb{Z}_p^*|}=[1]$ \u5373 $[a]^{p-1}=[1]$. $\\square$<\/p>\n
\u547d\u98981.3.6 (Euler \u5b9a\u7406) \u5982\u679c $(a,p)=1$ \u5219\u6709 $a^{\\varphi(p)}\\equiv 1\\pmod p$.<\/h4>\n
$\\varphi$ \u88ab\u79f0\u4e3a\u6b27\u62c9\u51fd\u6570, \u5176\u5b9a\u4e49\u662f $\\varphi :\\mathbb {Z^+}\\to \\mathbb {N}, p\\mapsto \\varphi(p):=|\\{a\\in \\mathbb {Z}|0<a<p\\land (a,p)=1\\}|$.<\/p>\n
Proof.<\/p>\n
\u56e0\u4e3a $(a,p)=1$, \u6240\u4ee5 $[a]\\in \\mathbb {Z}_p^*$.<\/p>\n
\u7531\u4e8e $\\mathbb {Z}_p^*$ \u5173\u4e8e\u5269\u4f59\u7c7b\u4e58\u6cd5\u6784\u6210\u7fa4, \u6240\u4ee5 $[a]^{|\\mathbb{Z}_p^*|}=[1]$ \u5373 $[a]^{\\varphi(p)}=[1]$. $\\square$<\/p>\n
\u547d\u98981.3.7 \u8bbe $H,K$ \u90fd\u662f\u7fa4 $G$ \u7684\u6709\u9650\u5b50\u7fa4, \u5219 $|HK|=\\frac {|H|\\cdot|K|} {|H\\cap K|}$.<\/h4>\n
Proof.<\/p>\n
$HK=\\bigcup_{h\\in H}hK=\\coprod_{A\\in\\{hK|h\\in H\\}}A$.<\/p>\n
\u6240\u4ee5 $|HK|=\\sum_{A\\in\\{hK|h\\in H\\}}|A|=\\sum_{A\\in\\{hK|h\\in H\\}}|K|=|\\{hK|h\\in H\\}|\\cdot|K|$.<\/p>\n
\u800c $\\forall h_1,h_2\\in H,(h_1K=h_2K\\Leftrightarrow h_1^{-1}h_2\\in K\\Leftrightarrow h_1^{-1}h_2\\in H\\cap K)$.<\/p>\n
\u6240\u4ee5 $|\\{hK|h\\in H\\}|=|\\{h(H\\cap K)|h\\in H\\}|=|(H\/(H\\cap K))_l|=[H:H\\cap K]=\\frac{|H|}{|H\\cap K|}$.<\/p>\n
\u6240\u4ee5 $|HK|=\\frac {|H|\\cdot|K|} {|H\\cap K|}$. $\\square$<\/p>\n
1.4 \u7fa4\u7684\u540c\u6001\u3001\u540c\u6784\u3001\u76f4\u79ef\u3001\u6b63\u89c4\u5b50\u7fa4\u3001\u5546\u7fa4<\/h3>\n\u5b9a\u4e491.4.1 \u540c\u6001\u6620\u5c04 (Homomorphism)<\/h4>\n
\u8bbe $G$ \u548c $G'$ \u662f\u4e24\u4e2a\u7fa4, \u5982\u679c\u6620\u5c04 $\\sigma:G\\to G'$ \u6ee1\u8db3 $\\sigma(ab)=\\sigma(a)\\sigma(b)$, \u5219\u79f0 $\\sigma$ \u662f\u4e00\u4e2a\u540c\u6001\u6620\u5c04<\/strong>.<\/p>\n\u5982\u679c $\\sigma$ \u662f\u5355\u5c04, \u5219\u79f0\u4e3a\u5355\u540c\u6001<\/strong> (injective homomorphism).<\/p>\n\u5982\u679c $\\sigma$ \u662f\u6ee1\u5c04, \u5219\u79f0\u4e3a\u6ee1\u540c\u6001<\/strong> (surjective homomorphism).<\/p>\n$\\mathrm {Im}(\\sigma):=\\{\\sigma(a)|a\\in G\\}$, \u88ab\u79f0\u4e3a $\\sigma$ \u7684\u50cf<\/strong> (image).<\/p>\n$\\mathrm {Ker}(\\sigma):=\\{a\\in G|\\sigma(a)=e'\\}$, \u88ab\u79f0\u4e3a $\\sigma$ \u7684\u6838<\/strong> (kernel). ($e'$ \u4e3a $G'$ \u7684\u5355\u4f4d\u5143)<\/p>\n\u547d\u98981.4.1 \u8bbe $\\sigma$ \u662f\u7fa4 $G$ \u5230\u7fa4 $G'$ \u7684\u540c\u6001\u6620\u5c04, \u5219 $\\mathrm {Im}(\\sigma)<G'$.-<\/h4>\n
Proof.<\/p>\n
\u4efb\u53d6 $a',b'\\in \\mathrm {Im}(\\sigma),\\exists a,b\\in G,\\sigma(a)=a',\\sigma(b)=b'$.<\/p>\n
\u56e0\u4e3a\u65b9\u7a0b $ax=b$ \u5728 $G$ \u4e2d\u6709\u89e3 $x_0$, \u6240\u4ee5 $\\sigma(ax_0)=\\sigma(b)\\Rightarrow \\sigma(a)\\sigma(x_0)=\\sigma(b)\\Rightarrow a'\\sigma(x_0)=b'$.<\/p>\n
\u6240\u4ee5\u65b9\u7a0b $a'x'=b'$ \u5728 $\\mathrm {Im}(\\sigma)$ \u4e2d\u6709\u89e3.<\/p>\n
\u540c\u7406\u5f97\u65b9\u7a0b $y'a'=b'$ \u5728 $\\mathrm {Im}(\\sigma)$ \u4e2d\u6709\u89e3.<\/p>\n
\u6240\u4ee5 $\\mathrm {Im}(\\sigma)$ \u5173\u4e8e $G'$ \u7684\u8fd0\u7b97\u6784\u6210\u7fa4, $\\mathrm {Im}(\\sigma)<G'$. $\\square$<\/p>\n
\u547d\u98981.4.2 \u8bbe $\\sigma$ \u662f\u7fa4 $G$ \u5230\u7fa4 $G'$ \u7684\u540c\u6001\u6620\u5c04, \u5219 $\\sigma(e)=e',\\sigma(a^{-1})=\\sigma(a)^{-1}$.<\/h4>\n
(\u5176\u4e2d $e$ \u4e3a $G$ \u7684\u5355\u4f4d\u5143, $e'$ \u4e3a $G'$ \u7684\u5355\u4f4d\u5143)<\/p>\n
Proof.<\/p>\n
\n- \n
\u5148\u8bc1 $\\sigma(e)=e'$.<\/p>\n
$\\forall a'\\in \\mathrm {Im}(\\sigma),\\exists a\\in G,\\sigma(a)=a'$.<\/p>\n
$\\sigma(e)a'=\\sigma(e)\\sigma(a)=\\sigma(ea)=\\sigma(a)=a'$, \u540c\u7406\u5f97 $a'\\sigma(e)=a'$<\/p>\n
\u6240\u4ee5 $\\sigma(e)$ \u662f\u7fa4 $\\mathrm {Im}(\\sigma)$ \u7684\u5355\u4f4d\u5143, \u53c8\u56e0\u4e3a $\\mathrm {Im}(\\sigma)<G$, \u6240\u4ee5 $\\sigma(e)=e'$.<\/p>\n<\/li>\n
- \n
\u518d\u8bc1 $\\sigma(a^{-1})=\\sigma(a)^{-1}$.<\/p>\n
$\\forall a\\in G,\\sigma(a^{-1})\\sigma(a)=\\sigma(a^{-1}a)=\\sigma(e)=e'$.<\/p>\n
\u540c\u7406\u5f97 $\\sigma(a)\\sigma(a^{-1})=e'$.<\/p>\n
\u6240\u4ee5 $\\sigma(a^{-1})=\\sigma(a)^{-1}$. $\\square$<\/p>\n<\/li>\n<\/ul>\n
\u547d\u98981.4.3 \u8bbe $\\sigma$ \u662f\u7fa4 $G$ \u5230\u7fa4 $G'$ \u7684\u540c\u6001\u6620\u5c04, \u5219 $\\sigma$ \u662f\u5355\u5c04\u5f53\u4e14\u4ec5\u5f53 $\\mathrm {Ker}(\\sigma)=\\{e\\}$.<\/h4>\n
Proof.<\/p>\n
\n- \n
($\\Rightarrow $)<\/p>\n
$\\forall a\\in \\mathrm {Ker}(\\sigma),\\sigma(a)=e'=\\sigma(e)\\Rightarrow a=e$.<\/p>\n<\/li>\n
- \n
($\\Leftarrow$)
\n$$ \\begin{align}
\n \\sigma(a)=\\sigma(b) &\\Rightarrow \\sigma(a)\\sigma(b^{-1})=\\sigma(b)\\sigma(b^{-1})\\\\
\n &\\Rightarrow \\sigma(ab^{-1})=e'\\\\
\n &\\Rightarrow ab^{-1}\\in \\mathrm {Ker}(\\sigma)=\\{e\\}\\\\
\n &\\Rightarrow ab^{-1}=e\\\\
\n &\\Rightarrow a=b\\ \\square
\n \\end{align}
\n$$<\/p>\n<\/li>\n<\/ul>\n
\u547d\u98981.4.4 \u8bbe $\\sigma$ \u662f\u7fa4 $G$ \u5230\u7fa4 $G'$ \u7684\u4e00\u4e2a\u5355\u540c\u6001, \u5219 $\\forall a\\in G,|a|=|\\sigma(a)|$.<\/h4>\n
Proof.<\/p>\n
\n- \n
\u82e5 $|a|<\\infty$<\/p>\n
\u5219\u6709 $e'=\\sigma(e)=\\sigma(a^{|a|})=\\sigma(a)^{|a|}$.<\/p>\n
\u6240\u4ee5 $\\left|\\sigma(a)\\right|\\mid |a|$.<\/p>\n
\u56e0\u4e3a $\\sigma$ \u662f\u5355\u540c\u6001, \u6240\u4ee5 $\\mathrm {Ker}(\\sigma)=\\{e\\}$.<\/p>\n
\u53c8\u6709 $e'=\\sigma(a)^{|\\sigma(a)|}=\\sigma(a^{|\\sigma(a)|})$.<\/p>\n
\u6240\u4ee5 $a^{|\\sigma(a)|}\\in \\mathrm {Ker}(\\sigma)\\Rightarrow a^{|\\sigma(a)|}=e\\Rightarrow |a|\\mid|\\sigma(a)|$.<\/p>\n
\u6240\u4ee5 $|a|=|\\sigma(a)|$.<\/p>\n<\/li>\n
- \n
\u82e5 $|a|=\\infty$<\/p>\n
\u5219 $\\forall n\\in \\mathbb {Z},a^n\\ne e$.<\/p>\n
\u6240\u4ee5 $a^n\\notin \\mathrm {Ker}(\\sigma)\\Rightarrow \\sigma(a)^n=\\sigma(a^n)\\ne e'$.<\/p>\n
\u6240\u4ee5 $|\\sigma(a)|=\\infty$. $\\square$<\/p>\n<\/li>\n<\/ul>\n
\u5b9a\u4e491.4.2 \u540c\u6784 (Isomorphism)<\/h4>\n
\u8bbe $G$ \u548c $G'$ \u662f\u4e24\u4e2a\u7fa4, \u5982\u679c\u5b58\u5728\u4e00\u4e2a $G$ \u5230 $G'$ \u7684\u53cc\u5c04\u662f\u540c\u6001\u6620\u5c04, \u5219\u79f0 $G$ \u548c $G'$ \u662f\u540c\u6784\u7684<\/strong> (isomorphic), \u8bb0\u4f5c $G\\cong G'$.<\/p>\n\u79f0 $\\sigma$ \u662f $G$ \u5230 $G'$ \u7684\u540c\u6784\u6620\u5c04<\/strong> (isomorphism), \u7b80\u79f0\u4e3a\u540c\u6784<\/strong>.<\/p>\n\u4e0b\u8bc1\u540c\u6784\u5173\u7cfb ($\\cong$) \u662f\u7b49\u4ef7\u5173\u7cfb:<\/p>\n
Proof.<\/p>\n
\n- \n
\u81ea\u53cd\u6027:<\/p>\n
\u53d6\u6052\u7b49\u6620\u5c04 $1_G$, \u6ee1\u8db3 $1_G(ab)=ab=1_G(a)1_G(b)$ \u4e14 $1_G$ \u662f\u53cc\u5c04, \u6240\u4ee5 $G\\cong G$<\/p>\n<\/li>\n
- \n
\u5bf9\u79f0\u6027:<\/p>\n
\u82e5 $G\\cong G'$, \u5219 $\\exists$ \u540c\u6784\u6620\u5c04 $\\sigma:G\\to G'$<\/p>\n
\u4efb\u53d6 $a',b'\\in G',\\exists a,b\\in G,\\sigma(a)=a',\\sigma(b)=b'$<\/p>\n
\u5219\u7531 $\\sigma(ab)=a'b'$ \u77e5 $\\sigma^{-1}(a'b')=ab$<\/p>\n
\u6240\u4ee5 $\\sigma^{-1}(a'b')=ab=\\sigma^{-1}(a')\\sigma^{-1}(b')$.<\/p>\n
\u53c8\u56e0\u4e3a $\\sigma^{-1}$ \u662f\u53cc\u5c04, \u6240\u4ee5 $\\sigma^{-1}$ \u662f $G'$ \u5230 $G$ \u7684\u540c\u6784\u6620\u5c04, $G'\\cong G$.<\/p>\n<\/li>\n
- \n
\u4f20\u9012\u6027:<\/p>\n
\u82e5 $G\\cong H,H\\cong K$, \u5219 $\\exists$ \u540c\u6784\u6620\u5c04 $\\sigma:G\\to H,\\pi:H \\to K$.<\/p>\n
\u5219 $\\sigma\\circ\\pi(ab)=\\sigma(\\pi(a)\\pi(b))=\\sigma\\circ\\pi(a)\\sigma\\circ\\pi(b)$.<\/p>\n
\u53c8\u56e0\u4e3a $\\sigma,\\pi$ \u5747\u4e3a\u53cc\u5c04, \u6240\u4ee5 $\\sigma\\circ \\pi$ \u4e5f\u4e3a\u53cc\u5c04, \u6240\u4ee5 $\\sigma\\circ\\pi$ \u662f $G$ \u5230 $K$ \u7684\u540c\u6784\u6620\u5c04, $G\\cong K$. $\\square$<\/p>\n<\/li>\n<\/ul>\n
\u547d\u98981.4.5 \u4efb\u610f\u4e00\u4e2a\u65e0\u9650\u5faa\u73af\u7fa4\u90fd\u4e0e $\\mathbb {Z}$ \u540c\u6784, \u4efb\u610f\u4e00\u4e2a $m$ \u9636\u5faa\u73af\u7fa4\u90fd\u4e0e $\\mathbb {Z}_m$ \u540c\u6784.<\/h4>\n
Proof.<\/p>\n
\n- \n
\u82e5 $G$ \u662f\u4e00\u4e2a\u65e0\u9650\u5faa\u73af\u7fa4 $\\xi$ \u662f $G$ \u7684\u4e00\u4e2a\u751f\u6210\u5143.
\n\u5efa\u7acb\u6620\u5c04 $\\sigma: \\mathbb {Z}\\to G,n\\mapsto \\xi^n$, \u5219\u6709<\/p>\n
\n- \n
\u5148\u8bc1\u660e $\\sigma$ \u662f\u5355\u5c04:
\n$\\sigma(n)=\\sigma(m)\\Rightarrow \\xi^n=\\xi^m\\Rightarrow \\xi^{n-m}=e$.
\n\u7531 $|\\xi|=\\infty$ \u77e5 $n-m=0$, \u6240\u4ee5 $n=m$.<\/p>\n<\/li>\n
- \n
\u518d\u8bc1\u660e $\\sigma$ \u662f\u6ee1\u5c04:<\/p>\n
$\\forall a\\in G$, \u56e0\u4e3a $G=\\left<\\xi\\right>$, \u6240\u4ee5 $\\exists n\\in\\mathbb {Z},\\xi^n=a$, \u6240\u4ee5 $\\sigma(n)=\\xi^n=a$.<\/p>\n<\/li>\n<\/ul>\n
\u6240\u4ee5 $\\sigma$ \u662f\u53cc\u5c04. <\/p>\n
\u56e0\u4e3a $\\sigma (n+m)=\\xi^{n+m}=\\xi^n\\xi^m=\\sigma(n)\\sigma(m)$, \u6240\u4ee5 $\\sigma$ \u662f\u540c\u6001\u6620\u5c04.<\/p>\n
\u7531\u4e0a\u53ef\u77e5 $\\sigma$ \u662f\u540c\u6784\u6620\u5c04, \u6240\u4ee5 $\\mathbb {Z}\\cong G$.<\/p>\n<\/li>\n
- \n
\u82e5 $G$ \u662f\u4e00\u4e2a $m$ \u9636\u751f\u6210\u7fa4, $\\xi$ \u662f $G$ \u7684\u4e00\u4e2a\u751f\u6210\u5143, \u5219 $|\\xi|=m$.<\/p>\n
\u5efa\u7acb\u6620\u5c04 $\\sigma: \\mathbb {Z}_m\\to G,[i]\\mapsto \\xi^i$, \u5219<\/p>\n
\u82e5 $[i]=[j]\\Rightarrow i=j+km\\ (\\exists k\\in \\mathbb {Z})\\Rightarrow \\xi^i=\\xi^{i+km}=\\xi^j$<\/p>\n
\u6240\u4ee5 $[i]=[j]\\Rightarrow \\sigma([i])=\\sigma([j])$, $\\sigma$ \u662f\u826f\u5b9a\u4e49\u7684.<\/p>\n
\n- \n
\u5148\u8bc1\u660e $\\sigma$ \u662f\u5355\u5c04:<\/p>\n
$\\sigma([i])=\\sigma([j])\\Rightarrow \\xi^i=\\xi^j\\Rightarrow \\xi^{i-j}=e$.<\/p>\n
\u6240\u4ee5 $m\\mid (i-j)\\Rightarrow j\\in [i]\\Rightarrow [i]=[j]$.<\/p>\n<\/li>\n
- \n
\u518d\u8bc1\u660e $\\sigma$ \u662f\u6ee1\u5c04:<\/p>\n
$\\forall a\\in G,\\exists i\\in \\mathbb {Z},\\xi^i=a$, \u5219\u6709 $\\sigma([i])=\\xi^i=a$.<\/p>\n<\/li>\n<\/ul>\n
\u6240\u4ee5 $\\sigma$ \u662f\u53cc\u5c04.<\/p>\n
\u56e0\u4e3a $\\sigma([i]+[j])=\\sigma([i+j])=\\xi^{i+j}=\\xi^i\\xi^j=\\sigma([i])\\sigma([j])$, \u6240\u4ee5 $\\sigma$ \u662f\u540c\u6001\u6620\u5c04.<\/p>\n
\u7531\u4e0a\u53ef\u77e5 $\\sigma$ \u662f\u540c\u6784\u6620\u5c04, \u6240\u4ee5 $\\mathbb {Z}_m\\cong G$. $\\square$<\/p>\n<\/li>\n<\/ul>\n
\u5b9a\u4e491.4.3 \u76f4\u79ef (Direct Product)<\/h4>\n
\u8bbe $G$ \u548c $G'$ \u662f\u4e24\u4e2a\u7fa4, \u5728 $G\\times G'$ \u4e0a\u5b9a\u4e49\u4e00\u4e2a\u8fd0\u7b97 ($\\cdot$):
\n$$(g_1,g_1')\\cdot(g_2,g_2'):=(g_1g_2,g_1'g_2')
\n$$
\n\u663e\u7136\u8fd9\u4e2a\u8fd0\u7b97\u6ee1\u8db3\u7ed3\u5408\u5f8b, \u6709\u5355\u4f4d\u5143 $(e,e')$, \u6bcf\u4e2a\u5143\u7d20 $(g,g')$ \u90fd\u6709\u9006\u5143 $(g^{-1},(g')^{-1})$. <\/p>\n
\u56e0\u6b64 $G\\times G'$ \u6784\u6210\u4e00\u4e2a\u7fa4, \u79f0\u5b83\u4e3a $G$ \u548c $G'$ \u7684\u76f4\u79ef<\/strong>, \u4ecd\u8bb0\u4f5c $G\\times G'$.<\/p>\n\u5982\u679c $G$ \u548c $G'$ \u4e0a\u7684\u8fd0\u7b97\u662f\u52a0\u6cd5, \u6709\u65f6\u4e5f\u79f0\u4e3a\u76f4\u548c<\/strong> (direct sum), \u8bb0\u4f5c $G\\oplus G'$.<\/p>\n\u5982\u679c $G$ \u548c $G'$ \u90fd\u662f\u6709\u9650\u7fa4, \u663e\u7136\u6709 $|G\\times G'|=|G||G'|$<\/p>\n
\u547d\u98981.4.6 $\\mathbb {Z}_m\\times \\mathbb {Z}_n$ \u662f\u5faa\u73af\u7fa4\u5f53\u4e14\u4ec5\u5f53 $(m,n)=1$, \u4ece\u800c $\\mathbb {Z}_m\\times \\mathbb {Z}_n\\cong \\mathbb {Z}_{mn}$ \u5f53\u4e14\u4ec5\u5f53 $(m,n)=1$.<\/h4>\n
Proof.<\/p>\n
\n- \n
($\\Rightarrow $)<\/p>\n
\u5047\u8bbe $(m,n)=d\\ne 1$, \u4ee4 $m=m'd,n=n'd$, \u5219\u6709 <\/p>\n
$\\forall([a],[b])\\in \\mathbb {Z}_m\\times \\mathbb {Z}_n, m'n'd([a],[b])=(m'n'd[a],m'n'd[b])=(n'm[a],m'n[b])=([0],[0])$<\/p>\n
\u6240\u4ee5 $|([a],[b])|\\le m'n'd< mn$ \u4e0e $\\mathbb {Z}_m\\times \\mathbb {Z}_n$ \u662f\u5faa\u73af\u7fa4\u4ea7\u751f\u77db\u76fe.<\/p>\n<\/li>\n
- \n
($\\Leftarrow$)<\/p>\n
\u56e0\u4e3a $([1],[0])+([0],[1])=([0],[1])+([1],[0])=([1],[1])$<\/p>\n
\u5e76\u4e14 $|([1],[0])|=m,|([0],[1])|=n,(n,m)=1$<\/p>\n
\u7531\u547d\u9898 1.1.5 \u77e5 $|([1],[1])|=mn$, \u53c8\u56e0\u4e3a $|\\mathbb {Z}_m\\times \\mathbb {Z}_n|=mn$<\/p>\n
\u6240\u4ee5 $\\mathbb {Z}_m\\times \\mathbb {Z}_n=\\left<([1],[1])\\right>$. $\\square$<\/p>\n<\/li>\n<\/ul>\n
\u547d\u98981.4.7<\/h4>\n
\u8bbe $G$ \u662f\u4e00\u4e2a\u7fa4, $H,K$ \u662f $G$ \u7684\u4e24\u4e2a\u5b50\u7fa4. \u5982\u679c:<\/p>\n
\n- \n
$G=HK$<\/p>\n<\/li>\n
- \n
$H\\cap K=\\{e\\}$<\/p>\n<\/li>\n
- \n
$\\forall h\\in H,\\forall k\\in K,hk=kh$.<\/p>\n<\/li>\n<\/ol>\n
\u5219\u6709 $G\\cong H\\times K$.<\/p>\n
Proof.<\/p>\n
\u5efa\u7acb\u6620\u5c04 $\\sigma:H\\times K\\to G,(h,k)\\mapsto hk$.<\/p>\n
\n- \n
\u5148\u8bc1 $\\sigma$ \u662f\u540c\u6001<\/p>\n
$\\sigma(h_1h_2,k_1k_2)=h_1h_2k_1k_2=h_1k_1h_2k_2=\\sigma(h_1,k_1)\\sigma(h_2,k_2)$.<\/p>\n<\/li>\n
- \n
\u518d\u8bc1 $\\sigma$ \u662f\u5355\u5c04<\/p>\n
\u82e5 $\\sigma(h,k)=e$ \u5219 $hk=e\\Rightarrow h=k^{-1}\\in H\\cap K=\\{e\\}$<\/p>\n
\u6240\u4ee5 $h=k=e\\Rightarrow \\mathrm{Ker}(\\sigma)=\\{(e,e)\\}$<\/p>\n
\u6240\u4ee5 $\\sigma$ \u662f\u5355\u540c\u6001.<\/p>\n<\/li>\n
- \n
\u518d\u8bc1 $\\sigma$ \u662f\u6ee1\u540c\u6001.<\/p>\n
$\\forall g\\in G=HK,\\exists h\\in H,k\\in K$, \u6ee1\u8db3 $hk=g$.<\/p>\n
\u6240\u4ee5 $\\sigma(h,k)=hk=g$.<\/p>\n<\/li>\n<\/ul>\n
\u6240\u4ee5 $\\sigma$ \u662f\u540c\u6784\u6620\u5c04 $G\\cong H\\times K$.<\/p>\n
\u5b9a\u4e491.4.4 \u5171\u8f6d\u5b50\u7fa4 (Conjugate Subgroup)<\/h4>\n
\u8bbe $G$ \u662f\u4e00\u4e2a\u7fa4, $H<G$, \u5bf9\u4e8e\u7ed9\u5b9a $g\\in G$, \u79f0 $gHg^{-1}$ \u662f $H$ \u7684\u4e00\u4e2a\u5171\u8f6d\u5b50\u7fa4<\/strong>.<\/p>\n\u4e0b\u8bc1 $gHg^{-1}$ \u662f $G$ \u7684\u5b50\u7fa4:<\/p>\n
Proof.<\/p>\n
$\\forall h_1,h_2\\in H,gh_1g^{-1}(gh_2g^{-1})^{-1}=gh_1g^{-1}gh_2^{-1}g^{-1}=gh_1h_2^{-1}g^{-1}\\in gHg^{-1}$<\/p>\n
\u6240\u4ee5 $gHg^{-1}<G$. $\\square$<\/p>\n
\u5b9a\u4e491.4.5 \u6b63\u89c4\u5b50\u7fa4 (Normal Subgroup)<\/h4>\n
\u8bbe $G$ \u662f\u4e00\u4e2a\u7fa4, $N<G$, \u5982\u679c $N$ \u7684\u6240\u6709\u5171\u8f6d\u5b50\u7fa4\u90fd\u7b49\u4e8e\u5176\u81ea\u8eab (\u5373 $\\forall g\\in G,gNg^{-1}=N$), \u5219\u79f0 $N$ \u662f $G$ \u7684\u6b63\u89c4\u5b50\u7fa4<\/strong>, \u8bb0\u4f5c $N\\lhd G$.<\/p>\n\u547d\u98981.4.8 $N$ \u662f $G$ \u7684\u6b63\u89c4\u5b50\u7fa4\u5f53\u4e14\u4ec5\u5f53 $gN=Ng(\\forall g\\in G)$.<\/h4>\n
Proof.<\/p>\n
\u4efb\u53d6 $g\\in G$, \u5219<\/p>\n
$gNg^{-1}=N\\Leftrightarrow (gNg^{-1})g=Ng\\Leftrightarrow gN(g^{-1}g)=Ng\\Leftrightarrow gN=Ng$. $\\square$<\/p>\n
\u547d\u98981.4.9 \u8bbe $\\sigma$ \u662f $G$ \u5230 $G'$ \u7684\u4e00\u4e2a\u540c\u6001, \u5219 $\\mathrm {Ker}(\\sigma)\\lhd G$.<\/h4>\n
Proof.<\/p>\n
$\\forall g\\in G,\\forall x\\in \\mathrm {Ker}(\\sigma)$ <\/p>\n
$\\sigma(gxg^{-1})=\\sigma(g)\\sigma(x)\\sigma(g^{-1})=\\sigma(g)\\sigma(g^{-1})=e'$.<\/p>\n
\u6240\u4ee5 $gxg^{-1}\\in \\mathrm {Ker}(\\sigma)$.<\/p>\n
\u6240\u4ee5 $g\\mathrm {Ker}(\\sigma)g^{-1}=\\mathrm {Ker}(\\sigma)$. $\\square$<\/p>\n
\u547d\u98981.4.10 \u8bbe $H$ \u662f $G$ \u7684\u5b50\u7fa4, \u82e5 $[G:H]=2$, \u5219 $H\\lhd G$.<\/h4>\n
Proof.<\/p>\n
\u4efb\u53d6 $a\\in G\\backslash H$, \u5219\u56e0\u4e3a $[G:H]=2$,<\/p>\n
\u6240\u4ee5 $G=H\\coprod aH=H\\coprod Ha$.<\/p>\n
\u6240\u4ee5 $aH=Ha$. $\\square$ <\/p>\n
\u5b9a\u4e491.4.6 \u5546\u7fa4 (Quotient group)<\/h4>\n
\u8bbe $G$ \u662f\u4e00\u4e2a\u7fa4, $N\\lhd G$, \u5219\u6709 $(G\/N)_l=(G\/N)_r$, \u8bb0\u4f5c $G\/N$, \u5728 $G\/N$ \u4e2d\u5b9a\u4e49\u8fd0\u7b97 ($\\cdot$):
\n$$(aN)\\cdot(bN):=abN
\n$$
\n\u82e5 $aN=bN,cN=dN$, \u5219 $b^{-1}a\\in N,(cN=dN\\Rightarrow Nc=Nd\\Rightarrow cd^{-1}\\in N)\\Rightarrow b^{-1}acd^{-1}\\in N$.<\/p>\n
\u53c8\u7531\u4e8e $N\\lhd G$, \u6240\u4ee5 $d^{-1}(b^{-1}acd^{-1})d\\in N\\Rightarrow (bd)^{-1}ac\\in N$.<\/p>\n
\u6240\u4ee5 $bdN=acN$, $\\cdot$ \u662f\u826f\u5b9a\u4e49\u7684.<\/p>\n
\u79f0 $(G\/N,\\cdot)$ \u4e3a\u7fa4 $G$ \u5bf9\u4e8e\u5b83\u7684\u6b63\u89c4\u5b50\u7fa4 $N$ \u7684\u5546\u7fa4<\/strong>. <\/p>\n\u5b9a\u4e491.4.7 \u81ea\u7136\u540c\u6001 (Natural Homomorphism)<\/h4>\n
\u8bbe $G$ \u662f\u4e00\u4e2a\u7fa4, $N\\lhd G$, \u79f0 $\\pi:G\\to G\/N,a\\mapsto aN$, \u4e3a\u81ea\u7136\u540c\u6001<\/strong>, \u6216\u8005\u6807\u51c6\u540c\u6001<\/strong> (canonical homomorphism).<\/p>\n\u5bb9\u6613\u9a8c\u8bc1 $\\pi$ \u662f\u6ee1\u540c\u6001.<\/p>\n
\u547d\u98981.4.11 (\u7fa4\u540c\u6001\u57fa\u672c\u5b9a\u7406) \u8bbe $\\sigma$ \u662f $G$ \u5230 $G'$ \u7684\u4e00\u4e2a\u540c\u6001, \u5219 $\\mathrm{Im}(\\sigma)\\cong G\/\\mathrm {Ker}(\\sigma)$.<\/h4>\n
Proof.<\/p>\n
\u5efa\u7acb\u6620\u5c04 $\\pi:G\/\\mathrm {Ker}(\\sigma)\\to \\mathrm {Im}(\\sigma),a\\mathrm {Ker}(\\sigma)\\mapsto \\sigma(a)$.<\/p>\n
\u82e5 $a\\mathrm {Ker}(\\sigma)=b\\mathrm {Ker}(\\sigma)$, \u5219\u6709 $a^{-1}b\\in \\mathrm {Ker}(\\sigma)$.<\/p>\n
\u6240\u4ee5 $\\pi(a\\mathrm {Ker}(\\sigma))=\\sigma(a)=\\sigma(a)\\sigma(a^{-1}b)=\\sigma(b)=\\pi(b\\mathrm {Ker}(\\sigma))$, $\\pi$ \u662f\u826f\u5b9a\u4e49\u7684.<\/p>\n
\n- \n
\u5148\u8bc1 $\\pi$ \u662f\u540c\u6001.<\/p>\n
$\\pi((a\\mathrm {Ker}(\\sigma))(b\\mathrm {Ker}(\\sigma)))=\\pi(ab\\mathrm {Ker}(\\sigma))=\\sigma(ab)=\\sigma(a)\\sigma(b)=\\pi(a\\mathrm {Ker}(\\sigma))\\pi(b\\mathrm {Ker}(\\sigma))$.<\/p>\n<\/li>\n
- \n
\u518d\u8bc1 $\\pi$ \u662f\u5355\u5c04.<\/p>\n
$\\forall a\\mathrm {Ker(\\sigma)}\\in \\mathrm {Ker}(\\pi),\\pi(a\\mathrm {Ker}(\\sigma))=e'\\Rightarrow \\sigma(a)=e'\\Rightarrow a\\in \\mathrm {Ker}(\\sigma)$.<\/p>\n
\u6240\u4ee5 $a\\mathrm {Ker}(\\sigma)=e\\mathrm {Ker}(\\sigma)\\Rightarrow \\mathrm {Ker}(\\pi)=\\{e\\mathrm {Ker}(\\sigma)\\}$.<\/p>\n
\u4ee5 $\\pi$ \u662f\u5355\u540c\u6001.<\/p>\n<\/li>\n
- \n
\u518d\u8bc1 $\\pi$ \u662f\u6ee1\u5c04.<\/p>\n
$\\forall a'\\in \\mathrm {Im}(\\sigma),\\exists a\\in G,\\sigma(a)=a'\\Rightarrow \\pi(a\\mathrm {Ker}(\\sigma))=\\sigma(a)=a'$.<\/p>\n<\/li>\n<\/ul>\n
\u7531\u4e0a\u53ef\u77e5, $\\pi$ \u4e3a\u540c\u6784, \u6240\u4ee5 $\\mathrm{Im}(\\sigma)\\cong G\/\\mathrm {Ker}(\\sigma)$. $\\square$<\/p>\n
\u547d\u98981.4.12 \u7b2c\u4e00\u7fa4\u540c\u6784\u5b9a\u7406<\/h4>\n
\u8bbe $G$ \u662f\u4e00\u4e2a\u7fa4, $H<G,N\\lhd G$, \u5219<\/p>\n
\n- $HN<G$<\/li>\n
- $H\\cap N\\lhd H$ \u4e14 $ H\/H\\cap N \\cong HN\/N$.<\/li>\n<\/ol>\n
Proof.<\/p>\n
\n- \n
\u4efb\u53d6 $h\\in H$ \u6709 $N\\lhd N\\Rightarrow hN=Nh$<\/p>\n
\u6240\u4ee5 $HN=NH$, \u7531\u547d\u9898 1.1.4 \u77e5 $HN<G$. <\/p>\n<\/li>\n
- \n
\u7531\u4e8e $HN<G$ \u4e14 $N\\lhd G$,\u6240\u4ee5 $N\\lhd HN$.<\/p>\n
\u5efa\u7acb\u6620\u5c04 $\\sigma:H\\to HN\/N,h\\mapsto hN$.<\/p>\n
\u6613\u5f97 $\\sigma$ \u662f\u540c\u6001\u6620\u5c04. <\/p>\n
$h\\in \\mathrm {Ker(\\sigma)}\\Leftrightarrow \\sigma(h)=hN=eN\\Leftrightarrow h\\in N\\cap H$.<\/p>\n
\u6240\u4ee5 $\\mathrm {Ker}(\\sigma)=N\\cap H\\xRightarrow{\u547d\u98981.4.9} N\\cap H\\lhd H$.<\/p>\n
$\\forall h\\in H,n\\in N$, \u6709 $hnN=hN$, \u6240\u4ee5 $\\sigma(h)=hN=hnN$.<\/p>\n
\u6240\u4ee5 $\\sigma$ \u662f\u6ee1\u540c\u6001, \u5373 $\\mathrm {Im}(\\sigma)=HN\/N$.<\/p>\n
\u7531\u7fa4\u540c\u6001\u57fa\u672c\u5b9a\u7406\u77e5 $\\mathrm {Im}(\\sigma)\\cong H\/\\mathrm {Ker}$, \u5373 $HN\/N\\cong H\/H\\cap N$. $\\square$<\/p>\n<\/li>\n<\/ol>\n
\u547d\u98981.4.13 (\u7b2c\u4e8c\u7fa4\u540c\u6784\u5b9a\u7406) \u8bbe $G$ \u662f\u4e00\u4e2a\u7fa4 $N\\lhd G$ \u4e14 $H\\lhd G$ \u4e14 $N \\lhd H$, \u5219 $H\/N\\lhd G\/N$ \u4e14 $(G\/N)\/(H\/N)\\cong G\/H$.<\/h4>\n
Proof.<\/p>\n
\u5efa\u7acb\u6620\u5c04 $\\sigma:G\/N\\to G\/H,aN\\mapsto aH$.<\/p>\n
$aN=bN\\Rightarrow b^{-1}a\\in N\\subset H\\Rightarrow aH=bH\\Rightarrow \\sigma(aN)=\\sigma(bN)$, \u6240\u4ee5 $\\sigma$ \u662f\u826f\u5b9a\u4e49\u7684.<\/p>\n
\u6613\u5f97 $\\sigma$ \u662f\u540c\u6001\u6620\u5c04.<\/p>\n
$aN\\in \\mathrm {Ker}(\\sigma)\\Leftrightarrow \\sigma(aN)=eH\\Leftrightarrow a\\in H\\Leftrightarrow aN\\in H\/N$, \u6240\u4ee5 $\\mathrm {Ker}(\\sigma)=H\/N$.<\/p>\n
$\\forall aH\\in G\/H,\\sigma(aN)=aH$, \u6240\u4ee5 $\\mathrm {Im}(\\sigma)=G\/H$.<\/p>\n
\u7531\u7fa4\u540c\u6001\u57fa\u672c\u5b9a\u7406\u77e5 $\\mathrm {Im}(\\sigma)\\cong (G\/N)\/\\mathrm {Ker}(\\sigma)$, \u5373 $G\/H\\cong (G\/N)\/(H\/N)$. $\\square$<\/p>\n
1.5 \u53ef\u89e3\u7fa4\u3001\u5355\u7fa4\u3001Jordan-H\u00f6lder \u5b9a\u7406<\/h3>\n\u5b9a\u4e491.5.1 \u6362\u4f4d\u5b50 (Commutator) \u4e0e\u6362\u4f4d\u5b50\u7fa4 (Commutator Group)<\/h4>\n
\u79f0 $xyx^{-1}y^{-1}$ \u4e3a $x$ \u4e0e $y$ \u7684\u6362\u4f4d\u5b50<\/strong>, \u8bb0\u4f5c $[x,y]$.<\/p>\n\u663e\u7136\u6709 $xy=yx\\Leftrightarrow xyx^{-1}y^{-1}=e$.<\/p>\n
\u79f0 $G$ \u7684\u6240\u6709\u6362\u4f4d\u5b50\u751f\u6210\u7684\u5b50\u7fa4\u4e3a $G$ \u7684\u6362\u4f4d\u5b50\u7fa4<\/strong>, \u4e5f\u53eb\u505a\u5bfc\u7fa4<\/strong> (derived group), \u8bb0\u4f5c $[G,G]$ \u6216 $G'$ ($G'$ \u5728\u672c\u6587\u4e2d\u6709\u65f6\u4e5f\u6307\u4efb\u610f\u4e00\u4e2a\u7fa4).<\/p>\n\u5373 $G':=\\left< \\{xyx^{-1}y^{-1}|x,y\\in G\\}\\right>$.<\/p>\n
\u547d\u98981.5.1 \u7fa4 $G$ \u4e3a Abel \u7fa4\u5f53\u4e14\u4ec5\u5f53 $G'=\\{e\\}$.<\/h4>\n
Proof.<\/p>\n
\n- \n
($\\Rightarrow$). <\/p>\n
$\\forall x,y\\in G,xyx^{-1}y^{-1}=e.$<\/p>\n<\/li>\n
- \n
($\\Leftarrow$).<\/p>\n
$\\forall x,y\\in G,xyx^{-1}y^{-1}\\in G'=\\{e\\}\\Rightarrow xyx^{-1}y^{-1}=e\\Rightarrow xy=yx$. $\\square$<\/p>\n<\/li>\n<\/ul>\n
\u547d\u98981.5.2 $G'\\lhd G$<\/h4>\n
Proof.<\/p>\n
\u4efb\u53d6 $a\\in G'$ \u6709 $a=[x_1,y_1][x_2,y_2]\\cdots[x_m,y_m]$.<\/p>\n
\u5219 $gag^{-1}=(g[x_1,y_1]g^{-1})(g[x_2,y_2]g^{-1})(g[x_m,y_m]g^{-1})=[gx_1g^{-1},gy_1g^{-1}]\\cdots[gx_mg^{-1},gy_mg^{-1}]\\in G'$.<\/p>\n
\u6240\u4ee5 $G'\\lhd G$. $\\square$<\/p>\n
\u547d\u98981.5.3 \u8bbe $\\sigma$ \u662f\u7fa4 $G$ \u5230 $\\tilde G$ \u7684\u540c\u6001, \u5219 $\\mathrm{Im}(\\sigma)$ \u662f Abel \u7fa4 $\\Leftrightarrow G'\\subset \\mathrm{Ker}(\\sigma)$.<\/h4>\n
Proof.<\/p>\n
\n- \n
($\\Leftarrow$).<\/p>\n
$\\operatorname{Im}(\\sigma)$ \u662f Abel \u7fa4 $\\Rightarrow$ $\\forall x,y\\in G,\\sigma(x)\\sigma(y)=\\sigma(y)\\sigma(x)\\Rightarrow \\sigma(xyx^{-1}y^{-1})=e$.<\/p>\n
\u6240\u4ee5 $xyx^{-1}y^{-1}\\in \\operatorname{Ker}(\\sigma)$.<\/p>\n
\u6240\u4ee5 $G'\\subset \\operatorname{Ker}(\\sigma)$.<\/p>\n<\/li>\n
- \n
($\\Rightarrow$).<\/p>\n
$G'\\subset\\operatorname{Ker}(\\sigma)\\Rightarrow \\sigma(xyx^{-1}y^{-1})=e\\Rightarrow \\sigma(x)\\sigma(y)=\\sigma(y)\\sigma(x)$.<\/p>\n
\u6240\u4ee5 $\\operatorname{Im}(\\sigma)$ \u662f Abel \u7fa4. $\\square$<\/p>\n<\/li>\n<\/ul>\n
\u547d\u98981.5.4 $G\/G'$ \u662f\u963f\u8d1d\u5c14\u7fa4.<\/h4>\n
$G\/G'$ \u88ab\u79f0\u4e3a\u628a $G$ "Abel \u5316" (abelianise).<\/p>\n
Proof.<\/p>\n
\u5efa\u7acb\u6620\u5c04 $\\sigma:G\\to G\/G',g\\mapsto gG'$, \u5219\u6709 $\\operatorname{Ker}(\\sigma)=G'\\subset G'$.<\/p>\n
\u6240\u4ee5 $\\operatorname{Im}(\\sigma)=G\/G'$ \u662f Abel \u7fa4. $\\square$ <\/p>\n
\u547d\u98981.5.5 \u8bbe $N\\lhd G$, \u5219 $G\/N$ \u4e3a Abel \u7fa4\u5f53\u4e14\u4ec5\u5f53 $G'\\subset N$.<\/h4>\n
\u7531\u8be5\u547d\u9898\u53ef\u4ee5\u770b\u51fa $G\/G'$ \u662f $G$ \u6700\u5927\u7684 Abel \u5546\u7fa4.<\/p>\n
Proof.<\/p>\n
\u5efa\u7acb\u6620\u5c04 $\\sigma:G\\to G\/N,g\\mapsto gN$.<\/p>\n
\u7531\u547d\u9898 1.5.3 \u5f97 $\\operatorname{Im}(\\sigma)=G\/N$ \u4e3a\u963f\u8d1d\u5c14\u7fa4\u5f53\u4e14\u4ec5\u5f53 $G'\\subset\\operatorname{Ker}(\\sigma)=N$. $\\square$<\/p>\n
\u5b9a\u4e491.5.2 \u5bfc\u7fa4\u5217 (Derived Groups Series)<\/h4>\n
\u8bbe $G$ \u662f\u4e00\u4e2a\u7fa4, \u5c06 $G'$ \u7684\u6362\u4f4d\u5b50\u7fa4\u8bb0\u4f5c $G^{(2)}$, \u5c06 $G^{(k)}(k\\ge 2)$ \u7684\u6362\u4f4d\u5b50\u7fa4\u8bb0\u4f5c $G^{(k+1)}$. \u5219\u5f97\u5230\u4e00\u4e2a $G$ \u7684\u9012\u964d\u5b50\u7fa4\u5217:
\n$$G\\rhd G' \\rhd G^{(2)}\\rhd\\cdots\\rhd G^{(k)}\\rhd G^{(k+1)}\\rhd\\cdots
\n$$
\n\u79f0\u5b83\u4e3a $G$ \u7684\u5bfc\u7fa4\u5217<\/strong>.<\/p>\n\u5b9a\u4e491.5.3 \u53ef\u89e3\u7fa4 (Solvable Group)<\/h4>\n
\u8bbe $G$ \u662f\u4e00\u4e2a\u7fa4, \u5982\u679c\u6709\u4e00\u4e2a\u6b63\u6574\u6570 $k$ \u4f7f\u5f97 $G^{(k)}=\\{e\\}$, \u5219\u79f0 $G$ \u4e3a\u53ef\u89e3\u7fa4, \u5426\u5219\u79f0 $G$ \u4e3a\u4e0d\u53ef\u89e3\u7fa4. <\/p>\n
\u547d\u98981.5.6 \u53ef\u89e3\u7fa4\u7684\u5546\u7fa4\u662f\u53ef\u89e3\u7fa4.<\/h4>\n1.6 \u7fa4\u5728\u96c6\u5408\u4e0a\u7684\u4f5c\u7528<\/h3>\n\u5b9a\u4e491.6.1 \u7fa4\u5728\u96c6\u5408\u4e0a\u7684\u4f5c\u7528 (Action)<\/h4>\n
\u8bbe $G$ \u662f\u4e00\u4e2a\u7fa4, $\\Omega$ \u662f\u4e00\u4e2a\u975e\u7a7a\u96c6\u5408. \u5982\u679c\u6620\u5c04 $\\sigma:G\\times \\Omega\\to \\Omega,(a,x)\\mapsto a\\circ x$ \u6ee1\u8db3:
\n$$\\begin{align}
\n(ab)\\circ x=a\\circ(b\\circ x),\\ \\ \\ &\\forall a,b\\in G,\\forall x\\in\\Omega
\n\\\\
\ne\\circ x=x,\\ \\ \\ &\\forall x\\in \\Omega
\n\\end{align}
\n$$
\n\u90a3\u4e48\u5c31\u79f0\u7fa4 $G$ \u5728\u96c6\u5408 $\\Omega$ \u4e0a\u6709\u4e00\u4e2a\u4f5c\u7528.<\/p>\n
\u547d\u98981.6.1 \u8bbe\u7fa4 $G$ \u5728\u96c6\u5408 $\\Omega$ \u4e0a\u6709\u4e00\u4e2a\u4f5c\u7528, \u4efb\u7ed9 $a\\in G$, \u4ee4 $\\psi(a)x:=a\\circ x,\\forall x\\in \\Omega$, \u5219 $\\psi:a\\mapsto \\psi(a)$ \u662f $G$ \u5230 $S_{\\Omega}$ \u7684\u4e00\u4e2a\u7fa4\u540c\u6001.<\/h4>\n
Proof.<\/p>\n
$\\forall a,b\\in G,\\forall x\\in \\Omega,$<\/p>\n
$\\psi(ab)x=(ab)\\circ x=a\\circ (b\\circ x)=a\\circ(\\psi(b)x)=\\psi(a)(\\psi(b)x)=(\\psi(a)\\psi(b))x$<\/p>\n
\u6240\u4ee5 $\\psi(ab)=\\psi(a)\\psi(b)$. $\\square$<\/p>\n
\u5b9a\u4e491.6.2 \u4f5c\u7528\u7684\u6838<\/h4>\n
\u5728\u547d\u9898 1.6.1 \u4e2d\u5b9a\u4e49\u7684\u540c\u6001 $\\psi$ \u7684\u6838\u79f0\u4e3a\u8fd9\u4e2a\u4f5c\u7528\u7684\u6838<\/strong>, \u4e8e\u662f\u6709:
\n$$\\begin{align}
\na\\in \\mathrm {Ker}(\\psi)&\\Leftrightarrow \\psi(a)=1_\\Omega\\\\
\n&\\Leftrightarrow \\psi(a)x=x,\\forall x\\in \\Omega\\\\
\n&\\Leftrightarrow a\\circ x=x,\\forall x\\in \\Omega
\n\\end{align}
\n$$
\n\u5f53 $\\mathrm {Ker}(\\psi)=\\{e\\}$ \u65f6, \u79f0\u8fd9\u4e2a\u4f5c\u7528\u662f\u5fe0\u5b9e\u7684<\/strong> (faithful), \u6b64\u65f6 $\\psi$ \u662f $G$ \u5230 $S_\\Omega$ \u7684\u4e00\u4e2a\u5355\u540c\u6001.<\/p>\n\u547d\u98981.6.2 \u8bbe\u7fa4 $G$ \u5230\u975e\u7a7a\u96c6\u5408 $\\Omega$ \u4e0a\u7684\u5168\u53d8\u6362\u7fa4\u6709\u4e00\u4e2a\u540c\u6001 $\\psi$, \u4ee4 $a\\circ x:=\\psi(a)x,\\forall a\\in G,\\forall x\\in \\Omega$, \u5219 $G$ \u5728 $\\Omega$ \u4e0a\u7531\u4e00\u4e2a\u4f5c\u7528 $(a,x)\\mapsto a\\circ x$.<\/h4>\n
Proof.<\/p>\n
\n- \n
\u5148\u8bc1\u660e $\\forall a,b\\in G,(ab)\\circ x=a\\circ(b\\circ x)$<\/p>\n
$(ab)\\circ x=\\psi(ab)x=(\\psi(a)\\psi(b))x=\\psi(a)(\\psi(b)x)=a\\circ(b\\circ x)$.<\/p>\n<\/li>\n
- \n
\u518d\u8bc1\u660e $e\\circ x=x$<\/p>\n
\u56e0\u4e3a $\\psi$ \u662f\u540c\u6001, \u6240\u4ee5 $\\psi(e)=1_\\Omega$ \u6240\u4ee5 $e\\circ x=1_\\Omega x=x$.<\/p>\n<\/li>\n<\/ul>\n
\u5b9a\u4e491.6.3 \u7fa4 $G$ \u5728\u96c6\u5408 $G$ \u4e0a\u7684\u5e73\u79fb<\/h4>\n
\u8bbe $G$ \u662f\u4e00\u4e2a\u7fa4\u4ee4 $(a,x)\\mapsto ax$ \u4e3a\u7fa4 $G$ \u5728\u96c6\u5408 $G$ \u4e0a\u7684\u4f5c\u7528, \u79f0\u5b83\u4e3a\u7fa4 $G$ \u5728\u96c6\u5408 $G$ \u4e0a\u7684\u5de6\u5e73\u79fb<\/strong> (left translation).<\/p>\n\u540c\u6837\u7684\u65b9\u5f0f\u8fd8\u53ef\u4ee5\u5b9a\u4e49\u7fa4 $G$ \u5728\u96c6\u5408 $G$ \u4e0a\u7684\u53f3\u5e73\u79fb.<\/p>\n
\u547d\u98981.6.3 (Cayley \u5b9a\u7406) \u4efb\u4f55\u4e00\u4e2a\u7fa4\u90fd\u540c\u6784\u4e0e\u67d0\u4e00\u96c6\u5408\u4e0a\u7684\u53d8\u6362\u7fa4.<\/h4>\n
Proof.<\/p>\n
\u8bbe $G$ \u662f\u4efb\u610f\u4e00\u4e2a\u7fa4, \u8003\u8651\u7fa4 $G$ \u5728\u96c6\u5408 $G$ \u4e0a\u7684\u5de6\u5e73\u79fb\u4f5c\u7528.<\/p>\n
\u7531 $ax=x\\Leftrightarrow a=e$ \u77e5\u8fd9\u4e2a\u4f5c\u7528\u7684\u6838\u4e3a $\\{e\\}$.<\/p>\n
\u4e8e\u662f\u5de6\u5e73\u79fb\u5f15\u8d77\u4e86\u7fa4 $G$ \u5230 $S_G$ \u7684\u4e00\u4e2a\u5355\u540c\u6001 $\\psi$, \u56e0\u6b64 $G\\cong G\/\\mathrm {Ker}(\\psi)\\cong \\mathrm {Im}(\\psi)$.<\/p>\n
\u800c $\\mathrm {Im}(\\psi)<S_G$, \u56e0\u6b64\u7fa4 $G$ \u4e0e\u96c6\u5408 $G$ \u4e0a\u7684\u4e00\u4e2a\u53d8\u6362\u7fa4\u540c\u6784. $\\square$<\/p>\n
\u63a8\u8bba: \u4efb\u4f55\u4e00\u4e2a\u6709\u9650\u7fa4\u90fd\u540c\u6784\u4e8e\u4e00\u4e2a\u7f6e\u6362\u7fa4. $\\square$<\/p>\n
\u5b9a\u4e491.6.4 \u7fa4 $G$ \u5728\u5de6\u5546\u96c6 $(G\/H)_l$ \u4e0a\u7684\u5de6\u5e73\u79fb<\/h4>\n
\u8bbe $H$ \u662f\u7fa4 $G$ \u7684\u4e00\u4e2a\u5b50\u7fa4, \u4ee4 $G\\times (G\/H)_l\\to (G\/H)_l,(a,xH)\\mapsto axH$.<\/p>\n
\u7531\u4e8e $(ab)\\circ xH=(ab)xH=a(bxH)=a\\circ(b\\circ xH),\\forall{a,b\\in H,\\forall xH\\in(G\/H)_l}$<\/p>\n
\u800c\u4e14 $e\\circ xH=exH=xH,\\forall xH\\in(G\/H)_l$,<\/p>\n
\u56e0\u6b64\u8fd9\u7ed9\u51fa\u4e86\u7fa4 $G$ \u5728\u5de6\u5546\u96c6\u4e0a\u7684\u4e00\u4e2a\u4f5c\u7528, \u79f0\u5b83\u4e3a\u7fa4 $G$ \u5728 $(G\/H)_l$ \u4e0a\u7684\u5de6\u5e73\u79fb<\/strong>.<\/p>\n\u7c7b\u4f3c\u7684, \u53ef\u4ee5\u8ba8\u8bba\u7fa4 $G$ \u5728\u53f3\u5546\u96c6 $(G\/H)_r$ \u4e0a\u7684\u53f3\u5e73\u79fb.<\/p>\n
\u5b9a\u4e491.6.5 \u5171\u8f6d\u4f5c\u7528 (Conjugation Action)<\/h4>\n
\u4ee4 $G\\times G\\to G,(a,x)\\mapsto axa^{-1}$.<\/p>\n
\u5bf9\u4e8e $\\forall a,b,x\\in G$, \u6709
\n$$\\begin{align}
\n(ab)\\circ x=(ab)x(ab)^{-1}&=a(bxb^{-1})a^{-1}=a\\circ(b\\circ x)\\\\
\nex&=exe^-1=x
\n\\end{align}
\n$$
\n\u6240\u4ee5\u8fd9\u7ed9\u51fa\u4e86\u7fa4 $G$ \u5728\u96c6\u5408 $G$ \u4e0a\u7684\u4e00\u4e2a\u4f5c\u7528, \u79f0\u5b83\u4e3a\u7fa4 $G$ \u5728\u96c6\u5408 $G$ \u4e0a\u7684\u5171\u8f6d\u4f5c\u7528.<\/p>\n
\u5b9a\u4e491.6.6 \u7fa4\u7684\u4e2d\u5fc3 (Centre)<\/h4>\n
$Z(G):=\\{a\\in G|ax=xa,\\forall x\\in G\\}$, \u88ab\u79f0\u4e3a\u7fa4 $G$ \u7684\u4e2d\u5fc3<\/strong>.<\/p>\n\u7531\u4e8e $axa^{-1}=x\\Leftrightarrow ax=xa$, \u6240\u4ee5\u7fa4 $G$ \u7684\u4e2d\u5fc3\u662f\u5171\u8f6d\u4f5c\u7528\u7684\u6838.<\/p>\n
\u5b9a\u4e491.6.7 \u81ea\u540c\u6784 (Automorphism) \u4e0e\u5185\u81ea\u540c\u6784 (Inner Automorphism)<\/h4>\n
\u7fa4 $G$ \u5230\u81ea\u8eab\u7684\u4e00\u4e2a\u540c\u6784\u6620\u5c04\u88ab\u79f0\u4e3a $G$ \u7684\u4e00\u4e2a\u81ea\u540c\u6784<\/strong>, $G$ \u7684\u6240\u6709\u81ea\u540c\u6784\u5173\u4e8e\u6620\u5c04\u7684\u4e58\u6cd5\u6784\u6210\u4e00\u4e2a\u7fa4, \u79f0\u5b83\u4e3a\u81ea\u540c\u6784\u7fa4<\/strong>, \u8bb0\u4f5c $\\mathrm {Aut}(G)$.<\/p>\n$\\sigma_a(x):=axa^{-1},\\forall x\\in G$, \u8fd9\u6837\u5b9a\u4e49\u7684 $\\sigma_a$ \u88ab\u79f0\u4e3a $G$ \u7684\u4e00\u4e2a\u5185\u81ea\u540c\u6784<\/strong>, \u6240\u6709\u7684\u5185\u81ea\u540c\u5173\u4e8e\u6620\u5c04\u4e58\u6cd5\u6784\u6210\u4e00\u4e2a\u7fa4, \u79f0\u5b83\u4e3a\u5185\u81ea\u540c\u6784\u7fa4<\/strong>, \u8bb0\u4f5c$\\mathrm {Inn}(G)$.<\/p>\n\u4e0b\u8bc1 $\\mathrm {Inn}(G)\\vartriangleleft\\mathrm {Aut}(G)$.<\/p>\n
Proof.<\/p>\n
\u4efb\u53d6 $\\sigma_a\\in \\mathrm {Inn}(G),\\tau\\in \\mathrm {Aut}(G)$.<\/p>\n
\u5219 $\\forall x\\in G$,
\n$$(\\tau\\sigma_a\\tau^{-1})x=\\tau(a(\\tau^{-1}x)a^{-1})=\\tau(a)\\tau(\\tau^{-1}x)\\tau(a^{-1})=\\tau(a)x\\tau(a)^{-1}=\\sigma_{\\tau(a)}x
\n$$
\n\u6240\u4ee5 $\\tau\\sigma_a\\tau^{-1}=\\sigma_{\\tau(a)}\\in\\mathrm {Inn}(G)$. $\\square$<\/p>\n
\u7531\u4e8e $\\mathrm {Ker}(\\sigma)=Z(G),\\mathrm {Im}(\\sigma)=\\mathrm {Inn(G)}$, \u7531\u7fa4\u540c\u6001\u57fa\u672c\u5b9a\u7406\u5f97 $G\/\\mathrm {Z}(G)\\cong \\mathrm {Inn}(G)$.<\/p>\n
\u5b9a\u4e491.6.8 \u8f68\u9053 (Orbit)<\/h4>\n
\u8bbe\u7fa4 $G$ \u5728 $\\Omega$ \u4e0a\u53c8\u4e00\u4e2a\u4f5c\u7528, \u5bf9\u4e8e $x\\in \\Omega$, $G(x):=\\{g\\circ x|g\\in G\\}$.<\/p>\n
\u79f0 $G(x)$ \u662f $x$ \u7684 $G$ – \u8f68\u9053<\/strong>.<\/p>\n\u6240\u6709\u8f68\u9053\u7ec4\u6210\u7684\u96c6\u5408\u7ed9\u51fa\u4e86 $\\Omega$ \u7684\u4e00\u4e2a\u5212\u5206.<\/p>\n
\u5728 $\\Omega$ \u4e2d\u89c4\u5b9a\u4e00\u4e2a\u4e8c\u5143\u5173\u7cfb:
\n$$x\\sim y\\stackrel{\\mathrm {def}}{\\Longleftrightarrow} \\exists g\\in G,y=g\\circ x.
\n$$
\n\u4e0b\u8bc1 $\\sim$ \u662f\u7b49\u4ef7\u5173\u7cfb:<\/p>\n
Proof.<\/p>\n
\n- \n
\u81ea\u53cd\u6027<\/p>\n
$e\\circ x=x\\Rightarrow x\\sim x$.<\/p>\n<\/li>\n
- \n
\u5bf9\u79f0\u6027<\/p>\n
$x\\sim y\\Rightarrow y=g\\circ x\\Rightarrow x=g^{-1}\\circ y\\Rightarrow y\\sim x$.<\/p>\n<\/li>\n
- \n
\u4f20\u9012\u6027<\/p>\n
$x\\sim y\\land y\\sim z\\Rightarrow y=g\\circ x,z=h\\circ y\\Rightarrow z=h\\circ (g\\circ x)=(hg)\\circ x\\Rightarrow x\\sim z$. $\\square$<\/p>\n<\/li>\n<\/ul>\n
\u5219\u4ee5 $x$ \u4e3a\u5e26\u8868\u5143\u7684\u7b49\u4ef7\u7c7b $[x]=G(x)$, \u6240\u6709\u8f68\u9053\u7ec4\u6210\u7684\u96c6\u5408\u7ed9\u51fa\u4e86 $\\Omega$ \u7684\u4e00\u4e2a\u5212\u5206. <\/p>\n
\u6240\u4ee5 $\\Omega=\\coprod_{i\\in I} G(x_i)$, \u5176\u4e2d $\\{x_i|i\\in I\\}$ \u79f0\u4e3a $\\Omega$ \u7684 $G$ – \u8f68\u9053\u7684\u5b8c\u5168\u4ee3\u8868\u7cfb<\/strong>. <\/p>\n\u5982\u679c\u7fa4 $G$ \u5728\u96c6\u5408 $\\Omega$ \u4e0a\u7684\u4f5c\u7528\u53ea\u6709\u4e00\u6761\u8f68\u9053, \u5373 $\\forall x,y\\in \\Omega,\\exists g\\in G$ \u4f7f\u5f97 $y=g\\circ x$, \u5219\u79f0 $G$ \u5728 $\\Omega$ \u4e0a\u7684\u4f5c\u7528\u662f\u4f20\u9012\u7684<\/strong> (transitive). \u6b64\u65f6\u79f0 $\\Omega$ \u662f\u7fa4 $G$ \u7684\u4e00\u4e2a\u9f50\u6027\u7a7a\u95f4<\/strong> (homogeneous space).<\/p>\n\u8003\u8651\u7fa4 $G$ \u5728\u96c6\u5408 $G$ \u4e0a\u7684\u5171\u8f6d\u4f5c\u7528, $x$ \u7684\u8f68\u9053\u4e3a $G(x)=\\{gxg^{-1}|g\\in G\\}$, \u79f0\u5176\u4e3a $x$ \u7684\u5171\u8f6d\u7c7b<\/strong> (conjugacy class). \u7ed9\u5b9a $x\\in G$, \u4efb\u53d6 $g\\in G$, $gxg^{-1}$ \u79f0\u4e3a $x$ \u7684\u5171\u8f6d\u5143<\/strong> (conjugate elements).<\/p>\n\u5b9a\u4e491.6.9 \u7c7b\u65b9\u7a0b (Class Equation)<\/h4>\n
\u79f0
\n$$|G|=|Z(G)|+\\sum_{j=1}^r|G(x_j)|
\n$$
\n\u4e3a\u6709\u9650\u7fa4 $G$ \u7684\u7c7b\u65b9\u7a0b<\/strong>. \u5176\u4e2d $G(x_j)$ \u662f $x_j$ \u7684\u5171\u8f6d\u7c7b, $\\{x_1,\\dots,x_r\\}$ \u662f $G$ \u7684\u975e\u4e2d\u5fc3\u5143\u7d20\u7684\u5171\u8f6d\u7c7b\u7684\u5b8c\u5168\u4ee3\u8868\u7cfb.<\/p>\n\u5b9a\u4e491.6.10 \u7a33\u5b9a\u5b50 (Stabilizer)<\/h4>\n
\u8bbe\u7fa4 $G$ \u5728\u96c6\u5408 $\\Omega$ \u4e0a\u6709\u4e00\u4e2a\u4f5c\u7528. \u7ed9\u5b9a $x\\in\\Omega$, \u4ee4 $G_x:=\\{g\\in G|g\\circ x=x\\}$.<\/p>\n
\u79f0 $G_x$ \u662f $x$ \u7684\u7a33\u5b9a\u5b50<\/strong>, \u5bb9\u6613\u9a8c\u8bc1 $G_x$ \u662f$G$ \u7684\u4e00\u4e2a\u5b50\u7fa4. \u56e0\u6b64\u4e5f\u79f0 $G_x$ \u662f $x$ \u7684\u7a33\u5b9a\u5b50\u7fa4<\/strong>.<\/p>\n\u5f53\u4f5c\u7528\u662f\u7fa4 $G$ \u5728\u96c6\u5408 $G$ \u4e0a\u7684\u5171\u8f6d\u4f5c\u7528, $x$ \u7684\u7a33\u5b9a\u5b50 $G_x=\\{g\\in G|gx=xg\\}$ \u88ab\u79f0\u4e3a $x$ \u5728 $G$ \u91cc\u7684\u4e2d\u5fc3\u5316\u5b50<\/strong> (centralizer), \u8bb0\u4f5c $C_G(x)$.<\/p>\n\u547d\u98981.6.4 (\u8f68\u9053-\u7a33\u5b9a\u5b50\u5b9a\u7406) \u8bbe\u7fa4 $G$ \u5728\u96c6\u5408 $\\Omega$ \u4e0a\u6709\u4e00\u4e2a\u4f5c\u7528, \u5219 $\\forall x\\in \\Omega$, \u6709 $|G(x)|=[G:G_x]$.<\/h4>\n
\u5373, $x$ \u7684\u8f68\u9053\u7684\u57fa\u6570\u7b49\u4e8e $x$ \u7684\u7a33\u5b9a\u5b50\u5728 $G$ \u4e2d\u7684\u6307\u6570.<\/p>\n
Proof.<\/p>\n
\u56e0\u4e3a $a\\circ x=b\\circ x \\Leftrightarrow b^{-1}\\circ (a\\circ x)=b^{-1}\\circ(b\\circ x)\\Leftrightarrow (b^{-1}a)\\circ x=x$,<\/p>\n
\u6240\u4ee5 $a\\circ x=b\\circ x \\Leftrightarrow b^{-1}a\\in G_x \\Leftrightarrow aG_x=bG_x$.<\/p>\n
\u5efa\u7acb\u6620\u5c04 $\\varphi: (G\/G_x)_l\\to G(x),aH\\mapsto a\\circ x$.<\/p>\n
\u7531 $aG_x=bG_x\\Rightarrow a\\circ x=b\\circ x$ \u53ef\u77e5 $\\varphi$ \u662f\u826f\u5b9a\u4e49\u7684.<\/p>\n
\u7531 $\\varphi(a_1G_x)=\\varphi(a_2G_x)\\Rightarrow a_1\\circ x=a_2\\circ x\\Rightarrow a_1G_x=a_2G_x$ \u77e5 $\\varphi$ \u662f\u5355\u5c04.<\/p>\n
\u7531 $\\forall a\\circ x\\in G_x,\\varphi(aG_x)=a\\circ x$ \u77e5 $\\varphi$ \u662f\u6ee1\u5c04.<\/p>\n
\u6240\u4ee5 $\\varphi$ \u662f\u53cc\u5c04, $|G(x)|=(G\/G_x)_l=[G:G_x]$. $\\square$<\/p>\n
\u63a8\u8bba: \u5982\u679c\u6709\u9650\u7fa4 $G$ \u5728\u96c6\u5408 $\\Omega$ \u4e0a\u53c8\u4e00\u4e2a\u4f5c\u7528, \u5219\u6bcf\u4e00\u6761\u8f68\u9053\u7684\u957f\u662f $G$ \u7684\u9636\u7684\u56e0\u5b50, \u5373 $|G|=|G_x||G(x)|$.<\/p>\n
\u5b9a\u4e491.6.11 \u4e0d\u52a8\u70b9 (Fixed Point)<\/h4>\n
\u8bbe\u7fa4 $G$ \u5728 $\\Omega$ \u4e0a\u53c8\u4e00\u4e2a\u4f5c\u7528. \u5bf9\u4e8e\u7ed9\u5b9a\u7684 $g$, \u82e5 $g\\circ x=x$, \u5219\u79f0 $x$ \u662f $g$ \u7684\u4e0d\u52a8\u70b9<\/strong>.<\/p>\n\u8bb0 $F(g):=\\{x\\in\\Omega|g\\circ x=x\\}$ \u4e3a $g$ \u7684\u4e0d\u52a8\u70b9\u96c6<\/strong>.<\/p>\n\u547d\u98981.6.5 (Burnside \u5f15\u7406) \u8bbe\u6709\u9650\u7fa4 $G$ \u5728\u6709\u9650\u96c6\u5408 $\\Omega$ \u4e0a\u6709\u4e00\u4e2a\u4f5c\u7528, \u5219\u8f68\u9053\u7684\u6761\u6570 $r=\\frac{1}{|G|}\\sum_{g\\in G}|F(g)|$.<\/h4>\n
Proof.<\/p>\n
\u8003\u8651 $S=\\{(g,x)\\in G\\times\\Omega|g\\circ x=x\\}$.<\/p>\n
\u5219\u6709 $G_x=\\{g\\in G|g\\circ x=x\\}=\\{g\\in G|(g,x)\\in S\\}$, $F(g)=\\{x\\in \\Omega|g\\circ x=x\\}=\\{x\\in \\Omega| (g,x)\\in S\\}$.<\/p>\n
\u6240\u4ee5 $\\sum_{x\\in \\Omega}|G_x|=|S|=\\sum_{g\\in G}|F(g)|$.<\/p>\n
\u7531\u8f68\u9053-\u7a33\u5b9a\u5b50\u5b9a\u7406\u77e5 $|G|=\\frac{1}{|I|}\\sum_{i\\in I}|G|=\\frac{1}{|I|}\\sum_{i\\in I}|G_{x_i}||G(x_i)|=\\frac{1}{r}\\sum_{i\\in I}|G_{x_i}||G(x_i)|$, \u5176\u4e2d $\\{x_1,x_2,\\dots,x_r\\}$ \u6784\u6210 $\\Omega$ \u7684 $G$ – \u8f68\u9053\u7684\u5b8c\u5168\u4ee3\u8868\u7cfb.
\n$$\\begin{align}
\n|G|&=\\frac{1}{r}\\sum_{i\\in I}|G_{x_i}||G(x_i)|\\\\
\n&=\\frac{1}{r}\\sum_{i\\in I}\\sum_{x\\in G(x_i)}|G_{x_i}|\\\\
\n&=\\frac{1}{r}\\sum_{i\\in I}\\sum_{x\\in G(x_i)}|G_{x}|\\\\
\n&=\\frac{1}{r}\\sum_{x\\in \\Omega}|G_{x}|\\\\
\n&=\\frac{1}{r}\\sum_{g\\in G} F(g)
\n\\end{align}
\n$$
\n\u6240\u4ee5 $r=\\frac{1}{|G|}\\sum_{g\\in G}|F(g)|$. $\\square$<\/p>\n
\u547d\u98981.6.6 (P\u00f3lya \u5b9a\u7406) \u8bbe\u6709\u9650\u7fa4 $G$ \u4f5c\u7528\u5728 $n$ \u4e2a\u5bf9\u8c61\u7ec4\u6210\u7684\u96c6\u5408 $W$ \u4e0a. $G$ \u4e2d\u5143\u7d20 $g$ \u5728 $W$ \u4e0a\u7684\u7f6e\u6362\u8868\u793a\u8bb0\u4f5c $\\tilde{g}$. \u7528 $m$ \u79cd\u989c\u8272\u7ed9 $W$ \u91cc\u7684 $n$ \u4e2a\u5bf9\u8c61\u67d3\u8272, \u5219\u771f\u6b63\u4e0d\u540c\u7684\u67d3\u8272\u65b9\u6848 $r=\\frac{1}{|G|}\\sum_{g\\in G}m^{r(\\tilde g)}$, \u5176\u4e2d $r(\\tilde g)$ \u662f $\\tilde g$ \u7684\u8f6e\u6362\u8868\u793a\u4e2d\u7684\u8f6e\u6362\u4e2a\u6570 (\u5305\u62ec $1$ – \u8f6e\u6362).<\/h4>\n
Proof.<\/p>\n
\u7531 Burnside \u5f15\u7406\u77e5 $r=\\frac{1}{|G|}\\sum_{g\\in G}|F(g)|=\\frac{1}{|G|}\\sum_{g\\in G} m^{r(\\tilde g)}$. $\\square$<\/p>\n","protected":false},"excerpt":{"rendered":"
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