群论(高等群论)

• 1. 前置
• 2. 同构定理(Isomorphism Theorems)
  • 2.1. 第一同构定理(First Isomorphism Theorem)
  • 2.2. 引理 1
  • 2.3. 引理 2
  • 2.4. 第二同构定理(Second Isomorphism Theorem)
  • 2.5. 第三同构定理(Third Isomorphism Theorem)
  • 2.6. 一些性质
• 3. Sylow Theorems
  • 3.1. Cauchy’s Theorem
  • 3.2. Sylow’s Theorems
  • 3.3. First Sylow Theorem
  • 3.4. Second Sylow Theorem
  • 3.5. Third Sylow Theorem

1. 前置

• 群论(二)

2. 同构定理(Isomorphism Theorems)

2.1. 第一同构定理(First Isomorphism Theorem)

设 $\varphi :G\to G^{\prime }$ 是群 $G$ 到 $G^{\prime }$ 的同态，则$G/\ker [\varphi ]\cong \varphi (G)$

2.2. 引理 1

$N◃G,\gamma :G\to G/N$为经典同态映射，$A=\{L◃G\mid N\le L\},B=\{L^{\prime }◃(G/N)\}$，则$\varphi :A\to B=\gamma [L]$为双射

2.3. 引理 2

$HN=\{hn\mid h\in H,n\in N\}$
$H\vee N=\bigcap \{E\le G\mid HN\subseteq E\}$,$H\vee N$为包含$HN$的最小子群

$N◃G,H\le G$,则$H\vee N=HN=NH$，且若$H◃G$，则$H\vee N=HN=NH◃G$

2.4. 第二同构定理(Second Isomorphism Theorem)

$N◃G,H\le G$,则$(HN)/N\cong H/(H\cap N)$

2.5. 第三同构定理(Third Isomorphism Theorem)

$K◃H◃G$,则$G/H\cong (G/K)/(H/K)$

2.6. 一些性质

$N◃G,H\le G\Rightarrow (H\cap N)◃H$

3. Sylow Theorems

Sylow Theorems部分的 $p$ 都是素数

let $X$ be a set and $G$ a group. A group action of $G$ on $X$ is a function $\varphi :G\times X\to X$ such that

将$\varphi (g,x)$记为$g\cdot x$或$gx$

• $e\cdot x=x$ for all $x\in X$
• $(gh)\cdot x=g\cdot (h\cdot x)$ for all $g,h\in G$ and $x\in X$

定义关系$x\sim y$当且仅当存在$g\in G$使得$y=g\cdot x$

$x$的等价类记作$[x]=:Gx=\{gx\mid g\in G\}$,也称作$x$的轨道(orbit)

$G_x=\{g\in G\mid g\cdot x=x\}$, 易得$G_x$是$G$的子群

$G\text{-}set$的一个结论：$|Gx|=|G:G_x|$(等势)

$X_G=\{x\in X\mid g\cdot x=x\text{ for all }g\in G\}$

若$X$为有限集，则

$$|X|=\sum _{x\in X}|Gx|$$ $$|X|=|X_G|+\sum _{x\in X\setminus X_G}|Gx|$$

Theorem. let $G$ be a group of order $p^n$ and let $X$ be a finite $G\text{-}set$. Then $|X|\equiv |X_G|(\mathrm{mod}p)$

$p\text{-}group$: let $p$ be a prime number. A group $G$ is called a $p\text{-}group$ if every element of $G$ has order a power of $p$

3.1. Cauchy’s Theorem

let $p$ be a prime number and $G$ a finite group and $p$ divides $|G|$. Then $G$ has an element of order $p$

Corollary. let $G$ be a finite group and $p$ a prime number. Then $G$ is a $p\text{-}group$ if and only if $|G|$ is a power of $p$

3.2. Sylow’s Theorems

$H\le G$

$N[H]=G_H=\{g\in G\mid gH{g}^{-1}=H\}$,易得$N[H]$是最大的以$H$为正规子群的$G$的子群，$N[H]$称作$H$在$G$中的正规化子(normalizer)

Lemma. Let H be a p-subgroup of a finite group G. Then $(N[H]:H)=(G:H)$

Corollary. let $H$ be a p-subgroup of finite group $G$. If p divides $(G:H)$, then $N[H]\ne H$

3.3. First Sylow Theorem

Let $G$ be a finite group and $p$ a prime number. if $|G|=p^{n}m$ where $p$ does not divide $m$, then

• $G$ contains a subgroup of order $p^i$ for each $i$ where $1\le i\le n$
• Every subgroup of order $p^i$ is normal in some subgroup of order $p^{i+1}$ for $1\le i<n$

3.4. Second Sylow Theorem

A Sylow $p$-subgroup of $G$ is a maximal $p$-subgroup of $G$, that is a $p$-subgroup contained no larger $p$-subgroup of $G$

Let $P_1$ and $P_2$ be Sylow $p$-subgroups of a finite group $G$. Then $P_1$ and $P_2$ are conjugate in $G$

3.5. Third Sylow Theorem

if $G$ is a finite group and $p$ is a prime number and $p$ divides $|G|$, then the number of Sylow $p$-subgroups of $G$ is congruent to $1$ modulo $p$ and divides $|G|$