# 问题求解4-2 群论初步

分类：Problem Solving， 发布于：2019-03-03 14:40:00， 更新于：2019-04-19 00:09:40。 评论

## 3.1 整数等价类和对称性

**Proposition 3.4.** Let

- Addition and multiplication are commutative:
$$a+b \equiv b+a \mod n$$ $$ab \equiv ba \mod n$$ - Addition and multiplication are associative:
$$(a+b)+c \equiv a+(b+c) \mod n$$ $$(ab)c \equiv a(bc) \mod n$$ - There are both additive and multiplicative identities:
$$a+0 \equiv a \mod n$$ $$a\cdot 0 \equiv 0 \mod n$$ - Multiplication distributes over addition:
$$a(b+c) \equiv ab+ac\mod n$$ - For every integer
$a$ there is an additive inverse$−a$ :$$a + (-a) \equiv 0 \mod n$$ - Let
$a$ be a nonzero integer. Then$\gcd(a, n) = 1$ if and only if there exists a multiplicative inverse$b$ for$a \mod n$ ; that is, a nonzero integer$b$ such that$$ab\equiv 1 \mod n$$

A **symmetry** of a geometric figure is a rearrangement of the figure preserving the arrangement of its sides and vertices as well as its distances and angles. A map from the plane to itself preserving the symmetry of an object is called a **rigid motion**.

## 3.2 定义与例子

### 群的定义

A **binary operation** or **law of composition** on a set

- The law of composition is
**associative**. That is,$$(a \circ b) \circ c = a \circ (b \circ c)$$ for$a, b, c \in G$ . - There exists an element
$e \in G$ , called the**identity element**, such that for any element$a \in G$ $$e \circ a = a \circ e = a$$ - For each element
$a \in G$ , there exists an**inverse element**in$G$ , denoted by$a^{-1}$ , such that$$a \circ a^{-1} = a^{-1} \circ a = e$$

A group **abelian（阿贝尔）** or **commutative**. Groups not satisfying this property are said to be **nonabelian** or **non-commutative**.

It is often convenient to describe a group in terms of an addition or multiplication table. Such a table is called a **Cayley table**.

A group is **finite**, or has **finite order**, if it contains a finite number of elements; otherwise, the group is said to be **infinite** or to have **infinite order**. The **order** of a finite group is the number of elements that it contains.

### 群的基本性质

**Proposition 3.17.** The identity element in a group

**Proposition 3.18.** If

**Proposition 3.19.** Let

**Proposition 3.20.** Let

**Proposition 3.21.** Let

**Proposition 3.22.** If

This proposition tells us that the right and left cancellation laws are true in groups.

**Theorem 3.23.** In a group, the usual laws of exponents hold; that is, for all

$g^m g^n = g^{m+n}$ for all$m, n \in \mathbb{Z}$ ;$(g^m)^n = g^{mn}$ for all$m, n \in \mathbb{Z}$ ;$(gh)^n = (h^{−1}g^{−1})^{−n}$ for all$n \in \mathbb{Z}$ . Furthermore, if$G$ is abelian, then$(gh)^n = g^n h^n$ .

## 3.3 子群

### 定义

We define a **subgroup**

The subgroup **trivial subgroup**. A subgroup that is a proper subset of **proper subgroup**.

### 定理

**Proposition 3.30.** A subset

- The identity
$e$ of$G$ is in$H$ . - If
$h_1, h_2 \in H$ , then$h_1 h_2 \in H$ . - If
$h \in H$ , then$h^{−1} \in H$ .

**Proposition 3.31.** Let

## 4.1 循环子群

### 循环群的定义

**Theorem 4.3.** Let

For **cyclic subgroup** generated by **cyclic group**. In this case **generator** of **order** of a to be the smallest positive integer **infinite** and write

**Theorem 4.9.** Every cyclic group is abelian.

### 循环群的子群

**Theorem 4.10.** Every subgroup of a cyclic group is cyclic.

**Corollary 4.11.** The subgroups of

**Proposition 4.12.** Let

**Theorem 4.13.** Let

**Corollary 4.14.** The generators of

## 4.2 复数乘法群

**Proposition 4.20.** Let

**Theorem 4.22 DeMoivre.** Let

### 圆群和单位根

We first consider **the circle group**,

**Proposition 4.24.** The circle group is a subgroup of

The complex numbers satisfying the equation **th roots of unity**.

**Theorem 4.25.** If

A generator for the group of the **primitive $n$th root of unity**.

## 4.3 快速幂

The laws of exponents still work in