# 问题求解4-4 群同态基本定理与正规子群

分类：Problem Solving， 发布于：2019-03-16 16:25:00， 更新于：2019-04-19 00:09:40。 评论

## 9.1 同构(isomorphism)的定义与例子

Two groups **isomorphic** if there exists a one-to-one and onto map
**isomorphism**.

**Theorem 9.6.** Let

$\phi^{-1}: H \rightarrow G$ is an isomorphism.$|G| = |H|$ .- If
$G$ is abelian, then$H$ is abelian. - If
$G$ is cyclic, then$H$ is cyclic. - If
$G$ has a subgroup of order$n$ , then$H$ has a subgroup of order$n$ .

**Theorem 9.7.** All cyclic groups of infinite order are isomorphic to

**Theorem 9.8.** If

**Corollary 9.9.** If

**Theorem 9.10.** The isomorphism of groups determines an equivalence relation on the class of all groups.

**Theorem 9.12 Cayley.**（凯莱定理） Every group is isomorphic to a group of permutations.

**left regular representation**:

## 9.2 直积

**Proposition 9.13.** Let

The group **external direct product** of

**Theorem 9.17.** Let

**Corollary 9.18.** Let

**Theorem 9.21.** The group

**Corollary 9.22** Let

**Corollary 9.23** If

Let

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

Then **internal direct product** of

**Theorem 9.27** Let

**Theorem 9.29** Let

## 10.1 商群（因子群）与正规子群

A subgroup **normal** in

**Theorem 10.3** Let

- The subgroup
$N$ is normal in$G$ . - For all
$g \in G$ ,$gNg^{-1} \subset N$ . - For all
$g \in G$ ,$gNg^{-1} = N$ .

If **factor** or **quotient group** of

**Theorem 10.4** Let

*It is very important to remember that the elements in a factor group are sets of elements in the original group.*

## 10.2 交错群的简单性

Groups with no nontrivial normal subgroups are called **simple groups**.

**Lemma 10.8** The alternating group

**Lemma 10.9** Let

**Lemma 10.10** For

**Theorem 10.11** The alternating group,

## 11.1 群同态(homomorphism)

A **homomorphism** between groups **homomorphic image** of

区别：同态不要求

**Proposition 11.4** Let

- If
$e$ is the identity of$G_1$ , then$\phi(e)$ is the identity of$G_2$ ; - For any element
$g \in G_1$ ,$\phi(g^{-1}) = [\phi(g)]^{-1}$ ; - If
$H_1$ is a subgroup of$G_1$ , then$\phi(H_1)$ is a subgroup of$G_2$ ; - If
$H_2$ is a subgroup of$G_2$ , then$\phi^{-1}(H_2) = \{g \in G_1: \phi(g) \in H_2\}$ is a subgroup of$G_1$ . Furthermore, if$H_2$ is normal in$G_2$ , then$\phi^{-1}(H_2)$ is normal in$G_1$ .

Let **kernel** of

**Theorem 11.5** Let

## 11.2 同构定理

Let **natural** or **canonical homomorphism**

The kerenel of this homomorphism is

**Theorem 11.10 First Isomorphism Theorem** If

**Theorem 11.12 Second Isomorphism Theorem** Let

**Theorem 11.13 Correspondence Theorem** Let

**Theorem 11.14 Third Isomorphism Theorem** Let