# 问题求解4-4 群同态基本定理与正规子群

## 9.1 同构(isomorphism)的定义与例子

Two groups $(G; \cdot)$ and $(H; \circ)$ are isomorphic if there exists a one-to-one and onto map $\phi: G\rightarrow H$ such that the group operation is preserved; that is, $$\phi(a\cdot b) = \phi(a) \circ \phi(b)$$ for all $a$ and $b$ in $G$. If $G$ is isomorphic to $H$, we write $G \cong H$ (\cong). The map $\phi$ is called an isomorphism.

Theorem 9.6. Let $\phi: G \rightarrow H$ be an isomorphism of two groups. Then the following statements are true.

1. $\phi^{-1}: H \rightarrow G$ is an isomorphism.
2. $|G| = |H|$.
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 $n$.

Theorem 9.7. All cyclic groups of infinite order are isomorphic to $\mathbb{Z}$.

Theorem 9.8. If $G$ is a cyclic group of order $n$, then $G$ is isomorphic to $\mathbb{Z}_n$.

Corollary 9.9. If $G$ is a group of order $p$, where $p$ is a prime number, then $G$ is isomorphic to $\mathbb{Z}_p$.

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: $\lambda_g: h \mapsto gh$

## 9.2 直积

Proposition 9.13. Let $G$ and $H$ be groups. The set $G\times H$ is a group under the operation $(g_1, h_1)(g_2, h_2) = (g_1g_2, h_1h_2)$ where $g_1, g_2 \in G$ and $h_1, h_2 \in H$.

The group $G\times H$ is called the external direct product of $G$ and $H$.

Theorem 9.17. Let $(g, h) \in G \times H$. If $g$ and $h$ have finite orders $r$ and $s$ respectively, then the order of $(g, h)$ in $G \times H$ is the least common multiple of $r$ and $s$.

Corollary 9.18. Let $(g_1, \dots, g_n) \in \prod G_i$. If $g_i$ has finite order $r_i$ in $G_i$, then the order of $(g_1, \dots, g_n)$ in $\prod G_i$ is the least common multiple of $r_1, \dots, r_n$.

Theorem 9.21. The group $\mathbb{Z}_m \times \mathbb{Z}_n$ is isomorphic to $\mathbb{Z}_{mn}$ if and only if $\gcd(m, n) = 1$.

Corollary 9.22 Let $n_1, \dots, n_k$ be positive integers. Then $$\prod\limits_{i=1}^k \mathbb{Z}_{n_i} \cong \mathbb{Z}_{n_1\cdots n_k}$$ if and only if $\gcd(n_i, n_j) = 1$ for $i \neq j$.

Corollary 9.23 If $$m=p_1^{e_1} \cdots p_k^{e_k},$$ where the $p_i$s are distinct primes, then $$\mathbb{Z}_m \cong \mathbb{Z}_{p_1^{e_1}} \times \cdots \times \mathbb{Z}_{p_k^{e_k}}.$$

Let $G$ be a group with subgroups $H$ and $K$ satisfying the following condifitons.

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

Then $G$ is the internal direct product of $H$ and $K$.

Theorem 9.27 Let $G$ be the internal direct product of subgroups $H$ and $K$. Then $G$ is isomorphic to $H \times K$.

Theorem 9.29 Let $G$ be the internal direct product of subgroups $H_i$, where $i = 1, 2, \dots, n$. Then $G$ is isomorphic to $\prod_i H_i$.

## 10.1 商群（因子群）与正规子群

A subgroup $H$ of a group $G$ is normal in $G$ if $gH=Hg$ for all $g \in G$.

Theorem 10.3 Let $G$ be a group and $N$ be a subgroup of $G$. Then the following statements are equivalent.

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

If $N$ is a normal subgroup of a group $G$, then the cosets of $N$ in $G$ form a group $G/N$ under the operation $(aN)(bN) = abN$. This group is called the factor or quotient group of $G$ and $N$.

Theorem 10.4 Let $N$ be a normal subgroup of a group $G$. The cosets of $N$ in $G$ form a group $G/N$ of order $[G : N]$.

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 $A_n$ is generated by 3-cycles for $n \geqslant 3$.

Lemma 10.9 Let $N$ be a normal subgroup of $A_n$, where $n \geqslant 3$. If $N$ contains a 3-cycle, then $N=A_n$.

Lemma 10.10 For $n \geqslant 5$, every nontrivial normal subgroup $N$ of $A_n$ contains a 3-cycle.

Theorem 10.11 The alternating group, $A_n$, is simple for $n \geqslant 5$.

## 11.1 群同态(homomorphism)

A homomorphism between groups $(G, \cdot)$ and $(H, \circ)$ is a map $\phi: G \rightarrow H$ such that $$\phi(g_1 \cdot g_2) = \phi(g_1) \circ \phi(g_2)$$ for $g_1, g_2 \in G$. The range of $\phi$ in $H$ is called the homomorphic image of $\phi$.

Proposition 11.4 Let $\phi: G_1 \rightarrow G_2$ be a homomorphism of groups. Then

1. If $e$ is the identity of $G_1$, then $\phi(e)$ is the identity of $G_2$;
2. For any element $g \in G_1$, $\phi(g^{-1}) = [\phi(g)]^{-1}$;
3. If $H_1$ is a subgroup of $G_1$, then $\phi(H_1)$ is a subgroup of $G_2$;
4. 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 $\phi: G \rightarrow H$ be a group homomorphism and suppose that $e$ is the identity of $H$. By Proposition 11.4, $\phi^{-1}(\{e\})$ is a subgroup of $G$. This subgroup is called the kernel of $\phi$ and will be denoted by $\ker \phi$.

Theorem 11.5 Let $\phi: G \rightarrow H$ be a group homomorphism. Then the kernel of $\phi$ is a normal subgroup of $G$.

## 11.2 同构定理

Let $H$ be a normal subgroup of $G$. Define the natural or canonical homomorphism $$\phi: G \rightarrow G/H$$ by $$\phi(g) = gH .$$

The kerenel of this homomorphism is $H$.

Theorem 11.10 First Isomorphism Theorem If $\psi: G \rightarrow H$ is a group homomorphism with $K = \ker \psi$, then $K$ is normal in $G$. Let $\phi: G \rightarrow G/K$ be the canonical homomorphism. Then there exists a unique isomorphism $\eta: G/K \rightarrow \psi(G)$ such that $\psi = \eta \phi$.

Theorem 11.12 Second Isomorphism Theorem Let $H$ be a subgroup of a group $G$ (not necessarily normal in $G$) and $N$ a nromal subgroup of $G$. Then $HN$ is a subgroup of $G$, $H \cap N$ is a normal subgroup of $H$, and $$H/H\cap N \cong HN/N$$

Theorem 11.13 Correspondence Theorem Let $N$ be a normal subgroup of a group $G$. Then $H \mapsto H/N$ is a one-to-one correspondence between the set of subgroups $H$ containing $N$ and the set of subgroups of $G/N$. Furthermore, the normal subgroups of $G$ containing $N$ correspond to normal subgroups of $G/N$.

Theorem 11.14 Third Isomorphism Theorem Let $G$ be a group and $N$ and $H$ be normal subgroups of $G$ with $N \subset H$. Then $$G/H \cong \dfrac{G/N}{H/N} .$$