schtonn
/
public


			
				
					
						
						
							12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849505152535455565758596061626364656667686970717273747576777879808182838485868788899091929394959697
							<!DOCTYPE html><html lang="en"><head><meta charset="UTF-8"><meta name="viewport" content="width=device-width,initial-scale=1,maximum-scale=2"><meta name="theme-color" content="#222"><meta name="generator" content="Hexo 4.2.0"><link rel="apple-touch-icon" sizes="180x180" href="/blog/blog/images/apple-touch-icon-next.png"><link rel="icon" type="image/png" sizes="32x32" href="/blog/blog/images/favicon-frog.png"><link rel="icon" type="image/png" sizes="16x16" href="/blog/blog/images/favicon-frog.png"><link rel="mask-icon" href="/blog/blog/images/logo.svg" color="#222"><link rel="stylesheet" href="/blog/css/main.css"><link rel="stylesheet" href="//fonts.googleapis.com/css?family=Comic Sans MS:300,300italic,400,400italic,700,700italic|Consolas:300,300italic,400,400italic,700,700italic&display=swap&subset=latin,latin-ext"><link rel="stylesheet" href="/blog/lib/font-awesome/css/font-awesome.min.css"><link rel="stylesheet" href="/blog/lib/pace/pace-theme-minimal.min.css"><script src="/blog/lib/pace/pace.min.js"></script><script id="hexo-configurations">var NexT=window.NexT||{},CONFIG={hostname:"schtonn.github.io",root:"/blog/",scheme:"Muse",version:"7.8.0",exturl:!1,sidebar:{position:"left",display:"post",padding:18,offset:12,onmobile:!1},copycode:{enable:!0,show_result:!0,style:"flat"},back2top:{enable:!0,sidebar:!1,scrollpercent:!0},bookmark:{enable:!1,color:"#222",save:"auto"},fancybox:!1,mediumzoom:!1,lazyload:!0,pangu:!1,comments:{style:"tabs",active:"valine",storage:!0,lazyload:!1,nav:null,activeClass:"valine"},algolia:{hits:{per_page:10},labels:{input_placeholder:"Search for Posts",hits_empty:"We didn't find any results for the search: ${query}",hits_stats:"${hits} results found in ${time} ms"}},localsearch:{enable:!1,trigger:"auto",top_n_per_article:1,unescape:!1,preload:!1},motion:{enable:!0,async:!1,transition:{post_block:"fadeIn",post_header:"slideDownIn",post_body:"slideDownIn",coll_header:"slideLeftIn",sidebar:"slideUpIn"}}}</script><meta name="description" content="定义
 定义 \((S,\cdot)\) 为一个群，其中 \(S\) 是一个非空集合，\(\cdot\) 是一个二元运算。
 则它满足以下四个条件：
 
 封闭性：\(\forall a,b\in S, a\cdot b\in S,\)
 结合律：\(\forall a,b,c\in S,(a\cdot b)\cdot c&#x3D;a\cdot(b\cdot c)\)
 单位元：\(\exists e\i"><meta property="og:type" content="article"><meta property="og:title" content="关于群论的一些东西"><meta property="og:url" content="https://schtonn.github.io/blog/posts/group/index.html"><meta property="og:site_name" content="schtonn"><meta property="og:description" content="定义
 定义 \((S,\cdot)\) 为一个群，其中 \(S\) 是一个非空集合，\(\cdot\) 是一个二元运算。
 则它满足以下四个条件：
 
 封闭性：\(\forall a,b\in S, a\cdot b\in S,\)
 结合律：\(\forall a,b,c\in S,(a\cdot b)\cdot c&#x3D;a\cdot(b\cdot c)\)
 单位元：\(\exists e\i"><meta property="og:locale" content="en_US"><meta property="article:published_time" content="2021-05-31T11:51:15.000Z"><meta property="article:modified_time" content="2022-10-19T15:33:01.165Z"><meta property="article:author" content="Alex"><meta property="article:tag" content="math"><meta name="twitter:card" content="summary"><link rel="canonical" href="https://schtonn.github.io/blog/posts/group/"><script id="page-configurations">CONFIG.page={sidebar:"",isHome:!1,isPost:!0,lang:"en"}</script><title>关于群论的一些东西 | schtonn</title><noscript><style>.sidebar-inner,.use-motion .brand,.use-motion .collection-header,.use-motion .comments,.use-motion .menu-item,.use-motion .pagination,.use-motion .post-block,.use-motion .post-body,.use-motion .post-header{opacity:initial}.use-motion .site-subtitle,.use-motion .site-title{opacity:initial;top:initial}.use-motion .logo-line-before i{left:initial}.use-motion .logo-line-after i{right:initial}</style></noscript></head><body itemscope itemtype="http://schema.org/WebPage"><div class="container use-motion"><div class="headband"></div><header class="header" itemscope itemtype="http://schema.org/WPHeader"><div class="header-inner"><div class="site-brand-container"><div class="site-nav-toggle"><div class="toggle" aria-label="Toggle navigation bar"><span class="toggle-line toggle-line-first"></span><span class="toggle-line toggle-line-middle"></span><span class="toggle-line toggle-line-last"></span></div></div><div class="site-meta"><a href="/blog/" class="brand" rel="start"><span class="logo-line-before"><i></i></span><h1 class="site-title">schtonn</h1><span class="logo-line-after"><i></i></span></a><p class="site-subtitle" itemprop="description">schtonn</p></div><div class="site-nav-right"><div class="toggle popup-trigger"></div></div></div><nav class="site-nav"><ul id="menu" class="menu"><li class="menu-item menu-item-home"><a href="/blog/" rel="section"><i class="fa fa-fw fa-home"></i> Home</a></li><li class="menu-item menu-item-tags"><a href="/blog/tags/" rel="section"><i class="fa fa-fw fa-tags"></i> Tags</a></li><li class="menu-item menu-item-archives"><a href="/blog/archives/" rel="section"><i class="fa fa-fw fa-archive"></i> Archives</a></li><li class="menu-item menu-item-games"><a href="/blog/games/" rel="section"><i class="fa fa-fw fa-gamepad"></i> Games</a></li></ul></nav></div></header><div class="back-to-top"><i class="fa fa-arrow-up"></i> <span>0%</span></div><main class="main"><div class="main-inner"><div class="content-wrap"><div class="content post posts-expand"><article itemscope itemtype="http://schema.org/Article" class="post-block" lang="en"><link itemprop="mainEntityOfPage" href="https://schtonn.github.io/blog/posts/group/"><span hidden itemprop="author" itemscope itemtype="http://schema.org/Person"><meta itemprop="image" content="/blog/images/avatar.gif"><meta itemprop="name" content="Alex"><meta itemprop="description" content="blog"></span><script type="text/javascript" src="/blog/js/md5.js"></script><script></script><script>document.oncopy=function(e){window.event&&(e=window.event);try{var t=e.srcElement;return"INPUT"==t.tagName&&"text"==t.type.toLowerCase()||"TEXTAREA"==t.tagName}catch(e){return!1}}</script><span hidden itemprop="publisher" itemscope itemtype="http://schema.org/Organization"><meta itemprop="name" content="schtonn"></span><header class="post-header"><h1 class="post-title" itemprop="name headline"> 关于群论的一些东西</h1><div class="post-meta"><span class="post-meta-item"><span class="post-meta-item-icon"><i class="fa fa-calendar-o"></i></span> <span class="post-meta-item-text">Posted on</span> <time title="Created: 2021-May-31 19:51:15" itemprop="dateCreated datePublished" datetime="2021-05-31T19:51:15+08:00">2021-May-31</time></span><span class="post-meta-item"><span class="post-meta-item-icon"><i class="fa fa-calendar-check-o"></i></span> <span class="post-meta-item-text">Edited on</span> <time title="Modified: 2022-Oct-19 23:33:01" itemprop="dateModified" datetime="2022-10-19T23:33:01+08:00">2022-Oct-19</time></span><span class="post-meta-item"><span class="post-meta-item-icon"><i class="fa fa-comment-o"></i></span> <span class="post-meta-item-text">Valine:</span><a title="valine" href="/blog/posts/group/#valine-comments" itemprop="discussionUrl"><span class="post-comments-count valine-comment-count" data-xid="/blog/posts/group/" itemprop="commentCount"></span></a></span></div></header><div class="post-body" itemprop="articleBody"><h2 id="定义">定义</h2><p>定义 <span class="math inline">\((S,\cdot)\)</span> 为一个群，其中 <span class="math inline">\(S\)</span> 是一个非空集合，<span class="math inline">\(\cdot\)</span> 是一个二元运算。</p><p>则它满足以下四个条件：</p><ol type="1"><li>封闭性：<span class="math inline">\(\forall a,b\in S, a\cdot b\in S,\)</span></li><li>结合律：<span class="math inline">\(\forall a,b,c\in S,(a\cdot b)\cdot c=a\cdot(b\cdot c)\)</span></li><li>单位元：<span class="math inline">\(\exists e\in S,\forall a \in S, a\cdot e=e\cdot a=a\)</span></li><li>逆元：<span class="math inline">\(\forall a\in S,\exists a^{-1}\in S, \text{s.t.}a\cdot a^{-1}=a^{-1}\cdot a=e\)</span></li></ol><a id="more"></a><p>半群满足以上条件1和2。</p><p>含幺半群满足条件1，2和3。</p><p>若 <span class="math inline">\((S,\cdot)\)</span> 是群，<span class="math inline">\(T\)</span> 是 <span class="math inline">\(S\)</span> 的非空子集，且 <span class="math inline">\((T,\cdot)\)</span> 也是群，则称 <span class="math inline">\((T,\cdot)\)</span> 是 <span class="math inline">\((S,\cdot)\)</span> 的子群。</p><p>注意，群不一定满足交换律。</p><p>左单位元：<span class="math inline">\(e_L\cdot a=a\)</span></p><p>右单位元：<span class="math inline">\(a\cdot e_R=a\)</span></p><p>左逆元：<span class="math inline">\(a_L^{-1}\cdot a=e\)</span></p><p>左逆元：<span class="math inline">\(a\cdot a_R^{-1}=e\)</span></p><p>置换群：有限集合到自身的一一映射称为一个置换。有限集合上的一些置换组成的集合，在置换的乘法下所组成的群，称为置换群。</p><p>二面体群：对平面上正 <span class="math inline">\(n\)</span> 边形所做的线性变换，包含 <span class="math inline">\(n\)</span> 个旋转 <span class="math inline">\(\rho_0\cdots\rho_{n-1}\)</span> 和 <span class="math inline">\(n\)</span> 个反射 <span class="math inline">\(\pi_0\cdots\pi_{n-1}\)</span>。</p><p>多面体群：对正多面体所做的线性变换。</p><p>子群：对于群 <span class="math inline">\((G,\cdot)\)</span>，</p><p>若 <span class="math inline">\(S\subseteq G,\text{s.t.}(S,\cdot)\)</span> 是群，则说 <span class="math inline">\((S,\cdot)\)</span> 是 <span class="math inline">\(G\)</span> 的子群，记为 <span class="math inline">\(S\leqslant G\)</span>。</p><p>当 <span class="math inline">\(S\neq G\)</span> 时，称为真子群。</p><h2 id="常见的群">常见的群</h2><p><span class="math inline">\((\mathbb{Z},+)\)</span>：整数加法；</p><p><span class="math inline">\((\mathbb{Q}^+,\times)\)</span>：正有理数乘法；</p><p><span class="math inline">\((\mathbb{Q}^-\cup\mathbb{Q}^-,\times)\)</span>：非零有理数乘法；</p><p><span class="math inline">\((\mathbb{Z}_n,+)\)</span>：<span class="math inline">\(\bmod n\)</span> 整数加法；</p><p>定义 <span class="math inline">\(\mathbb{Z}_n^*=\left\{x\mid (x,n)=1,1\leq x&lt;n\right\}\)</span></p><p><span class="math inline">\((\mathbb{Z}_n^*,\times)\)</span>：与 <span class="math inline">\(n\)</span> 互质的数的 <span class="math inline">\(\bmod n\)</span> 乘法；</p><p>正六面体变换群：<span class="math inline">\(D_6\)</span>。</p><p>Klein 四元素群：<span class="math inline">\(S=(\{e,a,b,c\},\cdot)\)</span>，其中 <span class="math inline">\(\forall x\in S,x\cdot x=e\)</span></p><h2 id="一些证明">一些证明</h2><p><span class="math inline">\(a\cdot b\)</span> 简写为 <span class="math inline">\(ab\)</span>。</p><h3 id="普遍群">普遍群</h3><h4 id="群中单位元唯一">群中单位元唯一</h4><p>设 <span class="math inline">\(e_1,e_2\)</span> 均为单位元。</p><p><span class="math display">\[e_1=e_1e_2=e_2\]</span></p><h4 id="群中不同元素的逆元各不相同">群中不同元素的逆元各不相同</h4><p>若 <span class="math inline">\(a\neq b,a^{-1}=b^{-1}\)</span></p><p>则</p><p><span class="math display">\[ab^{-1}=aa^{-1}=e\]</span> <span class="math display">\[ab^{-1}b=eb\]</span> <span class="math display">\[ae=be\]</span> <span class="math display">\[a=b\]</span></p><p>矛盾。</p><h4 id="若逆元都为自身则运算结果不同于被运算元素">若逆元都为自身，则运算结果不同于被运算元素</h4><p>若 <span class="math inline">\(\forall p\in S, p^{-1}=p\)</span></p><p>那么有 <span class="math inline">\(ab\neq b\)</span></p><p>假设 <span class="math inline">\(ab=b\)</span></p><p>则</p><p><span class="math display">\[(ab)b^{-1}=bb^{-1}\]</span> <span class="math display">\[ae=e\]</span> <span class="math display">\[a=e\]</span></p><p>矛盾。</p><h4 id="若半群包含左单位元和左逆元那么它是群">若半群包含左单位元和左逆元，那么它是群</h4><p>有半群 <span class="math inline">\((S,\cdot)\)</span>，使得：</p><ol type="1"><li><span class="math inline">\(\exists e_L,\forall a\in S,e_L\cdot a=a\)</span></li><li><span class="math inline">\(\forall a\in S,\exists a_L^{-1}\in S,\text{s.t.}a_L^{-1}\cdot a=e_L\)</span></li></ol><p>则 <span class="math inline">\(\forall a\in S\)</span></p><p><span class="math display">\[a\cdot a_L^{-1}\]</span></p><p><span class="math display">\[=e_L\cdot(a\cdot a_L^{-1})\]</span></p><p><span class="math display">\[=(a_L^{-1})_L^{-1}\cdot a_L^{-1}\cdot a\cdot a_L^{-1}\]</span></p><p><span class="math display">\[=(a_L^{-1})_L^{-1}\cdot(a_L^{-1}\cdot a)\cdot a_L^{-1}\]</span></p><p><span class="math display">\[=e_L\]</span></p><p>另 <span class="math inline">\(\forall a\in S\)</span></p><p><span class="math display">\[a\cdot e_L=a\cdot a_L^{-1}\cdot a=e_L\cdot a=a\]</span></p><p><span class="math display">\[\therefore e_L=e_R,a_L^{-1}=a_R^{-1}\]</span></p><p><span class="math inline">\(\therefore (S,\cdot)\)</span> 是群</p><h4 id="半群的以下方程有解是群的充要条件">半群的以下方程有解是群的充要条件</h4><p>有半群 <span class="math inline">\((S,\cdot)\)</span>，使得：</p><p><span class="math inline">\(\forall a,b\in S,ax=b,ya=b\)</span> 均有解</p><p><span class="math inline">\(\rArr\)</span> 若 <span class="math inline">\((S,\cdot)\)</span> 是群：</p><p><span class="math display">\[ax=b\]</span> <span class="math display">\[a^{-1}ax=a^{-1}b\]</span> <span class="math display">\[x=a^{-1}b\]</span></p><p>同理 <span class="math inline">\(y=ba^{-1}\)</span>。</p><p><span class="math inline">\(\lArr\)</span> 若方程有解：</p><p>取 <span class="math inline">\(b=a\)</span>，则有</p><p><span class="math display">\[ax=a,ya=a\]</span> <span class="math display">\[x=e_L,y=e_R\]</span></p><p>需证明 <span class="math inline">\(e_L\)</span> 和 <span class="math inline">\(e_R\)</span> 是单位元。</p><p><span class="math inline">\(\forall c\in S\)</span>：</p><p>令 <span class="math inline">\(qa=c,qae_R=qa=c\)</span></p><p>令 <span class="math inline">\(ap=c,e_Lap=ap=c\)</span></p><p><span class="math inline">\(\therefore e=e_L=e_R\)</span></p><p><span class="math inline">\(\forall d\in S\)</span></p><p>解 <span class="math inline">\(xd=e\)</span></p><p>则 <span class="math inline">\(x=d^{-1}\)</span>。</p><p><span class="math inline">\(\therefore\forall a,\exists a^{-1}\)</span></p><p><span class="math inline">\(\therefore (S,\cdot)\)</span> 是群</p><h4 id="有限半群的消去律成立是群的充要条件">有限半群的消去律成立是群的充要条件</h4><p>有有限半群 <span class="math inline">\((S,\cdot)\)</span>，使得：</p><ol type="1"><li><span class="math inline">\(ax=ay\)</span> 可推出 <span class="math inline">\(x=y\)</span>；</li><li><span class="math inline">\(xa=ya\)</span> 可推出 <span class="math inline">\(x=y\)</span>；</li></ol><p><span class="math inline">\(\rArr\)</span> 若 <span class="math inline">\((S,\cdot)\)</span> 是群：</p><p><span class="math display">\[xaa^{-1}=yaa^{-1}\]</span> <span class="math display">\[x=y\]</span></p><p><span class="math inline">\(ax=ay\)</span> 同理。</p><p><span class="math inline">\(\lArr\)</span> 若对任意元素消去律均成立：</p><p>设 <span class="math inline">\(S=\{a_1,a_2,\cdots,a_n\}\)</span></p><p>定义 <span class="math inline">\(S^\prime=\{a_1a,a_2a,\cdots,a_na\}\)</span></p><p>若 <span class="math inline">\(a_ia=a_ja\)</span>，则 <span class="math inline">\(a_i=a_j\)</span></p><p>若 <span class="math inline">\(a_ia\neq a_ja\)</span>，则 <span class="math inline">\(a_i\neq a_j\)</span></p><p><span class="math inline">\(\therefore S\)</span> 与 <span class="math inline">\(S^\prime\)</span> 一一对应。</p><p><span class="math display">\[\forall a_i \in S, \exists a_j\in S, \text{s.t.}a_ja=a_i\]</span></p><p><span class="math display">\[\forall a_i \in S, \exists a_j\in S, \text{s.t.}aa_j=a_i\]</span></p><p>由上一条证明得知，<span class="math inline">\(S\)</span> 是群。</p><h4 id="与n互质的数的mod-n乘法是群">与n互质的数的mod n乘法是群</h4><p>定义 <span class="math inline">\(\mathbb{Z}_n^*=\left\{x\mid (x,n)=1,1\leq x&lt;n\right\}\)</span></p><p><span class="math inline">\((\mathbb{Z}_n^*,\times)\)</span>：与 <span class="math inline">\(n\)</span> 互质的数的 <span class="math inline">\(\bmod n\)</span> 乘法是群。</p><p><span class="math display">\[\forall a,b\in\mathbb{Z}_n^*\]</span></p><p><span class="math display">\[\because(a,n)=1\]</span></p><p><span class="math display">\[\therefore\exists x_1,y_1\in\mathbb{Z},\text{s.t.}ax_1+ny_1=1\tag{1}\]</span></p><p><span class="math display">\[\because(b,n)=1\]</span></p><p><span class="math display">\[\therefore\exists x_2,y_2\in\mathbb{Z},\text{s.t.}bx_2+ny_2=1\tag{2}\]</span></p><p><span class="math display">\[(1)\times b:ax_1b+ny_1b=b\tag{3}\]</span></p><p><span class="math inline">\((3)\)</span> 代入 <span class="math inline">\((2)\)</span></p><p><span class="math display">\[ax_1bx_2+ny_1bx_2+ny_2=1\]</span></p><p><span class="math display">\[(ab)(x_1x_2)+n(y_1bx_2+y_2)=1\]</span></p><p><span class="math display">\[\therefore (ab,n)=1\]</span></p><p>封闭性成立。</p><p>结合律显然，半群成立。</p><p>消去律显然，群成立。</p><h3 id="子群">子群</h3><h4 id="以下三条等价">以下三条等价</h4><p>若 <span class="math inline">\((G,\cdot)\)</span> 是群，且 <span class="math inline">\(S\subseteq G\)</span> ，则</p><p><span class="math display">\[S\leqslant G\tag{1}\]</span></p><p><span class="math display">\[\forall a,b\in S,ab\in S,a^{-1}\in S\tag{2}\]</span></p><p><span class="math display">\[\forall a,b\in S,ab^{-1}\in S\tag{3}\]</span></p><p>三条等价。</p><p>证明：</p><p><span class="math inline">\((1)\Rightarrow(2)\)</span> 由定义。</p><p><span class="math inline">\((2)\Rightarrow(3)\)</span>：<span class="math inline">\(\forall a,b\in S,b^{-1}\in S,ab^{-1}\in S\)</span>。</p><p><span class="math inline">\((3)\Rightarrow(1)\)</span>：</p><p>取 <span class="math inline">\(a=b\)</span>。</p><p><span class="math display">\[aa^{-1}\in S\]</span> <span class="math display">\[e\in S\]</span></p><p>存在单位元。</p><p>取 <span class="math inline">\(a=e\)</span>。</p><p><span class="math display">\[\forall b\in S\]</span> <span class="math display">\[eb^{-1}\in S\]</span> <span class="math display">\[b^{-1}\in S\]</span></p><p>存在逆元。</p><p><span class="math display">\[\forall a,c\in S\]</span></p><p>令 <span class="math inline">\(b=c^{-1}\)</span>。</p><p><span class="math display">\[ab^{-1}\in S\]</span> <span class="math display">\[a(c^{-1})^{-1}\in S\]</span> <span class="math display">\[ac\in S\]</span></p><p>封闭性成立。</p><p>结合律显然，<span class="math inline">\((S,\cdot)\)</span> 是群</p><p><span class="math display">\[\therefore S\leqslant G\]</span></p><h4 id="子群的交也是子群">子群的交也是子群</h4><p>若 <span class="math inline">\(H_1,H_2\leqslant G\)</span>，则 <span class="math inline">\(H_1\cap H_2\leqslant G\)</span></p><p>证：</p><p><span class="math display">\[\forall a,b\in(H_1\cap H_2)\]</span></p><p><span class="math display">\[ab^{-1}\in H_1,ab^{-1}\in H_2\]</span></p><p><span class="math display">\[\therefore ab^{-1}\in(H_1\cap H_2)\]</span></p><p><span class="math display">\[\therefore H_1\cap H_2\leqslant G\]</span></p><h4 id="两个子群的并也是子群当且仅当一个子群是另一个子群的子群">两个子群的并也是子群，当且仅当一个子群是另一个子群的子群</h4><p>若 <span class="math inline">\(H_1\leqslant G,H_2\leqslant G\)</span>，</p><p>则 <span class="math inline">\(H_1\cup H_2\leqslant G\Leftrightarrow H_1\leqslant H_2\)</span> 或 <span class="math inline">\(H_2\leqslant H_1\)</span>。</p><p>证：</p><p><span class="math inline">\(\Leftarrow\)</span> 若 <span class="math inline">\(H_1\leqslant H_2\)</span>：</p><p>则 <span class="math inline">\(H_1\cup H_2=H_2\leqslant G\)</span>。</p><p><span class="math inline">\(\Rightarrow\)</span> 若 <span class="math inline">\(H_1\cap H_2\leqslant G\)</span></p><p>反设 <span class="math inline">\(H_1\nleqslant H_2\)</span> 且 <span class="math inline">\(H_2\nleqslant H_1\)</span></p><p><span class="math display">\[\therefore H_1-H_2\neq\phi,H_2-H_1\neq\phi\]</span></p><p>取 <span class="math inline">\(a\in H_1-H_2,b\in H_2-H_1\)</span></p><p>令 <span class="math inline">\(c=a\cdot b\)</span></p><p>若 <span class="math inline">\(c\in H_1\)</span>，则 <span class="math inline">\(\because a\in H_1,\therefore b=a^{-1}\cdot c\in H_1\)</span>，矛盾；</p><p>若 <span class="math inline">\(c\in H_2\)</span>，则 <span class="math inline">\(\because b\in H_2,\therefore a=c\cdot b^{-1}\in H_2\)</span>，矛盾。</p><p><span class="math inline">\(\therefore H_1\cap H_2\)</span> 不封闭，矛盾。</p><h4 id="两个子群间的运算具有交换律是将其合并成一个子群的充要条件">两个子群间的运算具有交换律是将其合并成一个子群的充要条件</h4><p>若 <span class="math inline">\(H_1,H_2\leqslant G\)</span></p><p>则 <span class="math inline">\(H_1H_2\leqslant G\Leftrightarrow H_1H_2=H_2H_1\)</span></p><p>其中 <span class="math inline">\(H_1H_2=\{ab\mid a\in H_1,b\in H_2\}\)</span></p><p><span class="math inline">\(\Rightarrow\)</span>：</p></div><div><ul class="post-copyright"><li class="post-copyright-author"> <strong>Post author:</strong> Alex</li><li class="post-copyright-link"> <strong>Post link:</strong> <a href="https://schtonn.github.io/blog/posts/group/" title="关于群论的一些东西">https://schtonn.github.io/blog/posts/group/</a></li><li class="post-copyright-license"> <strong>Copyright Notice:</strong> All articles in this blog are licensed under<a href="https://creativecommons.org/licenses/by-nc-sa/4.0/" rel="noopener" target="_blank"><i class="fa fa-fw fa-creative-commons"></i> BY-NC-SA</a> unless stating additionally.</li></ul></div><footer class="post-footer"><div class="post-tags"><a href="/blog/tags/math/" rel="tag"><i class="fa fa-tag"></i> math</a></div><div class="post-nav"><div class="post-nav-item"><a href="/blog/posts/matrix-pow-ext/" rel="prev" title="矩阵快速幂扩展应用"><i class="fa fa-chevron-left"></i> 矩阵快速幂扩展应用</a></div><div class="post-nav-item"> <a href="/blog/posts/powersum/" rel="next" title="自然数幂求和公式">自然数幂求和公式<i class="fa fa-chevron-right"></i></a></div></div></footer></article></div><div class="comments" id="valine-comments"></div><script>
  window.addEventListener('tabs:register', () => {
    let { activeClass } = CONFIG.comments;
    if (CONFIG.comments.storage) {
      activeClass = localStorage.getItem('comments_active') || activeClass;
    }
    if (activeClass) {
      let activeTab = document.querySelector(`a[href="#comment-${activeClass}"]`);
      if (activeTab) {
        activeTab.click();
      }
    }
  });
  if (CONFIG.comments.storage) {
    window.addEventListener('tabs:click', event => {
      if (!event.target.matches('.tabs-comment .tab-content .tab-pane')) return;
      let commentClass = event.target.classList[1];
      localStorage.setItem('comments_active', commentClass);
    });
  }
</script></div><div class="toggle sidebar-toggle"><span class="toggle-line toggle-line-first"></span><span class="toggle-line toggle-line-middle"></span><span class="toggle-line toggle-line-last"></span></div><aside class="sidebar"><div class="sidebar-inner"><ul class="sidebar-nav motion-element"><li class="sidebar-nav-toc"> Table of Contents</li><li class="sidebar-nav-overview"> Overview</li></ul><div class="post-toc-wrap sidebar-panel"><div class="post-toc motion-element"><ol class="nav"><li class="nav-item nav-level-2"><a class="nav-link" href="#定义"><span class="nav-number">1.</span> <span class="nav-text">定义</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="#常见的群"><span class="nav-number">2.</span> <span class="nav-text">常见的群</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="#一些证明"><span class="nav-number">3.</span> <span class="nav-text">一些证明</span></a><ol class="nav-child"><li class="nav-item nav-level-3"><a class="nav-link" href="#普遍群"><span class="nav-number">3.1.</span> <span class="nav-text">普遍群</span></a><ol class="nav-child"><li class="nav-item nav-level-4"><a class="nav-link" href="#群中单位元唯一"><span class="nav-number">3.1.1.</span> <span class="nav-text">群中单位元唯一</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#群中不同元素的逆元各不相同"><span class="nav-number">3.1.2.</span> <span class="nav-text">群中不同元素的逆元各不相同</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#若逆元都为自身则运算结果不同于被运算元素"><span class="nav-number">3.1.3.</span> <span class="nav-text">若逆元都为自身，则运算结果不同于被运算元素</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#若半群包含左单位元和左逆元那么它是群"><span class="nav-number">3.1.4.</span> <span class="nav-text">若半群包含左单位元和左逆元，那么它是群</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#半群的以下方程有解是群的充要条件"><span class="nav-number">3.1.5.</span> <span class="nav-text">半群的以下方程有解是群的充要条件</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#有限半群的消去律成立是群的充要条件"><span class="nav-number">3.1.6.</span> <span class="nav-text">有限半群的消去律成立是群的充要条件</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#与n互质的数的mod-n乘法是群"><span class="nav-number">3.1.7.</span> <span class="nav-text">与n互质的数的mod n乘法是群</span></a></li></ol></li><li class="nav-item nav-level-3"><a class="nav-link" href="#子群"><span class="nav-number">3.2.</span> <span class="nav-text">子群</span></a><ol class="nav-child"><li class="nav-item nav-level-4"><a class="nav-link" href="#以下三条等价"><span class="nav-number">3.2.1.</span> <span class="nav-text">以下三条等价</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#子群的交也是子群"><span class="nav-number">3.2.2.</span> <span class="nav-text">子群的交也是子群</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#两个子群的并也是子群当且仅当一个子群是另一个子群的子群"><span class="nav-number">3.2.3.</span> <span class="nav-text">两个子群的并也是子群，当且仅当一个子群是另一个子群的子群</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#两个子群间的运算具有交换律是将其合并成一个子群的充要条件"><span class="nav-number">3.2.4.</span> <span class="nav-text">两个子群间的运算具有交换律是将其合并成一个子群的充要条件</span></a></li></ol></li></ol></li></ol></div></div><div class="site-overview-wrap sidebar-panel"><div class="site-author motion-element" itemprop="author" itemscope itemtype="http://schema.org/Person"><p class="site-author-name" itemprop="name">Alex</p><div class="site-description" itemprop="description">blog</div></div><div class="site-state-wrap motion-element"><nav class="site-state"><div class="site-state-item site-state-posts"> <a href="/blog/archives"><span class="site-state-item-count">35</span> <span class="site-state-item-name">posts</span></a></div><div class="site-state-item site-state-tags"> <a href="/blog/tags/"><span class="site-state-item-count">8</span> <span class="site-state-item-name">tags</span></a></div></nav></div><div class="links-of-author motion-element"><span class="links-of-author-item"><a href="https://github.com/schtonn" title="GitHub → https:&#x2F;&#x2F;github.com&#x2F;schtonn" rel="noopener" target="_blank"><i class="fa fa-fw fa-github"></i> GitHub</a></span><span class="links-of-author-item"><a href="mailto:schtonn@163.com" title="E-Mail → mailto:schtonn@163.com" rel="noopener" target="_blank"><i class="fa fa-fw fa-envelope"></i> E-Mail</a></span></div><div class="links-of-blogroll motion-element"><div class="links-of-blogroll-title"><i class="fa fa-fw fa-link"></i> Links</div><ul class="links-of-blogroll-list"><li class="links-of-blogroll-item"> <a href="https://yonghong.github.io/" title="https:&#x2F;&#x2F;yonghong.github.io" rel="noopener" target="_blank">Yonghong</a></li><li class="links-of-blogroll-item"> <a href="https://source.unsplash.com/random/1600x900" title="https:&#x2F;&#x2F;source.unsplash.com&#x2F;random&#x2F;1600x900" rel="noopener" target="_blank">Background</a></li></ul><iframe width="400" height="300" frameborder="0" src="https://cdn.abowman.com/widgets/treefrog/index.html?up_bodyColor=2d2d2d&up_pattern=0&up_patternColor=000000&up_footColor=2d2d2d&up_eyeColor=3a3a3a&up_bellySize=50&up_backgroundColor=222222&up_tongueColor=2b2d2d&up_flyColor=3a3a3a&up_releaseFly=0"></iframe></div></div></div></aside><div id="sidebar-dimmer"></div></div></main><footer class="footer"><div class="footer-inner"><div class="copyright"> &copy; 2019 – <span itemprop="copyrightYear">2023</span><span class="with-love"><i class="fa fa-user"></i></span> <span class="author" itemprop="copyrightHolder">Alexander</span></div></div></footer></div><script src="/blog/lib/anime.min.js"></script><script src="//cdn.jsdelivr.net/npm/lozad@1/dist/lozad.min.js"></script><script src="/blog/lib/velocity/velocity.min.js"></script><script src="/blog/lib/velocity/velocity.ui.min.js"></script><script src="/blog/js/utils.js"></script><script src="/blog/js/motion.js"></script><script src="/blog/js/schemes/muse.js"></script><script src="/blog/js/next-boot.js"></script><script>!function(){var t=document.createElement("script"),e=window.location.protocol.split(":")[0];t.src="https"===e?"https://zz.bdstatic.com/linksubmit/push.js":"http://push.zhanzhang.baidu.com/push.js";var s=document.getElementsByTagName("script")[0];s.parentNode.insertBefore(t,s)}()</script><script>
  if (typeof MathJax === 'undefined') {
    window.MathJax = {
      loader: {
          load: ['[tex]/mhchem'],
        source: {
          '[tex]/amsCd': '[tex]/amscd',
          '[tex]/AMScd': '[tex]/amscd'
        }
      },
      tex: {
        inlineMath: {'[+]': [['$', '$']]},
          packages: {'[+]': ['mhchem']},
        tags: 'ams'
      },
      options: {
        renderActions: {
          findScript: [10, doc => {
            document.querySelectorAll('script[type^="math/tex"]').forEach(node => {
              const display = !!node.type.match(/; *mode=display/);
              const math = new doc.options.MathItem(node.textContent, doc.inputJax[0], display);
              const text = document.createTextNode('');
              node.parentNode.replaceChild(text, node);
              math.start = {node: text, delim: '', n: 0};
              math.end = {node: text, delim: '', n: 0};
              doc.math.push(math);
            });
          }, '', false],
          insertedScript: [200, () => {
            document.querySelectorAll('mjx-container').forEach(node => {
              let target = node.parentNode;
              if (target.nodeName.toLowerCase() === 'li') {
                target.parentNode.classList.add('has-jax');
              }
            });
          }, '', false]
        }
      }
    };
    (function () {
      var script = document.createElement('script');
      script.src = '//cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js';
      script.defer = true;
      document.head.appendChild(script);
    })();
  } else {
    MathJax.startup.document.state(0);
    MathJax.texReset();
    MathJax.typeset();
  }
</script><script>
NexT.utils.loadComments(document.querySelector('#valine-comments'), () => {
  NexT.utils.getScript('https://cdn.jsdelivr.net/npm/valine@1/dist/Valine.min.js', () => {
    var GUEST = ['nick', 'mail', 'link'];
    var guest = 'nick,mail';
    guest = guest.split(',').filter(item => {
      return GUEST.includes(item);
    });
    new Valine({
      el         : '#valine-comments',
      verify     : false,
      notify     : false,
      appId      : 'BmologYYnRqCv0SLHDeDdA17-gzGzoHsz',
      appKey     : 'w9mVebFMdCmY6Nh9vfcBGaGt',
      placeholder: "Comment...",
      avatar     : 'mp',
      meta       : guest,
      pageSize   : '10' || 10,
      visitor    : false,
      lang       : 'en' || 'zh-cn',
      path       : location.pathname,
      recordIP   : true,
      serverURLs : ''
    });
  }, window.Valine);
});
</script></body></html>