Mengzelev's Blog

# 问求学习笔记-群同构基本定理与正规子群

Word count: 1,687 / Reading time: 8 min
2019/03/14 Share

# 同构(Isomorphisms)

## 定义

$$\phi(a\cdot b)=\phi(a)\circ\phi(b)$$

## 基本定理

1. $\phi^{-1}:H\to G$ is an isomorphism (废话)
2. $|G|=|H|$ (废话)
3. If $G$ is abelian, then $H$ is abelian.
4. If $G$ is cyclic, then $H$ is cyclic.
5. If $G$ has a subgroup of order $n$, then $H$ has a subgroup of order $n$.

## Cayley’s Theorem

The isomorphism $g\mapsto \lambda_g$ is known as the left regular representationof G.

# 直积(Direct Products)

qlz管它叫直和

## 内直积(Internal Direct Product)

• $G=HK={hk:h\in H, k\in K}$
• $H\cap K={e}$
• $hk=kh$ for all $k\in K$ and $h\in H$

• $G=H_1H_2…H_n={h_1h_2…h_n:h_i\in H_i}$
• $H_i\cap\left\langle \cup_{j\neq i}H_j\right\rangle={e}$
• $H_ih_j=h_jh_i$ for all $h_i\in H_i$ and $h_j\in H_j$

# 正规子群(Normal Subgroups)

1. $N$是$G$的正规子群
2. $\forall g\in G, gNg^{-1}\subseteq N$.
3. $\forall g\in G, gNg^{-1}=N$.

# 商群(Factor Group)

$eN=N$是单位元，$g^{-1}N$是$gN$的逆元。

# 同态(homomorphism)

$\phi$在$H$中的值域被称为同态像(homomorphism image).

1. 若$e$是$G_1$的单位元，则$e$是$G_2$的单位元
2. 对于任意$g\in G_1$, $\phi(g^{-1})=[\phi(g)]^{-1}$
3. 若$H_1$是$G_1$的子群，则$\phi(H_2)$是$G_2$的子群
4. 若$H_2$是$G_2$的子群，则$\phi^{-1}(H_2)={g\in G_1: \phi(g)\in H_2}$是$G_1$的子群。此外，若$H_2$是$G_2$的正规子群，则$\phi^{-1}(H_2)$是$G_1$的正规子群

# 同态定理

$$H/H\cap N\cong HN/N$$