# 群和子群

## **群的定义**

> 非空集合$$G$$满足
>
> * **G1: 结合律**  $$\forall a,b,c\in G,\left(ab\right)c=a\left(bc\right)$$
> * **G2: 单位元**  $$\exists e\in G,\forall a\in G,ae=ea=a$$
> * **G3: 可逆性**  $$\forall a\in G,\exists a^{-1}\in G,aa^{-1}=a^{-1}a=e$$

## **Abel群/交换群**

群$$G$$满足

* **G4: 交换律**  $$\forall a,b\in G,ab=ba$$

## **定义和性质**

* 群$$G$$的元素个数叫做群$$G$$的阶，记作$$\left|G\right|$$
* 单位元唯一
* 逆元唯一
* $${\left(a\_1a\_2\cdots a\_n\right)}^{-1}=a\_{n}^{-1}\cdots a\_{2}^{-1}a\_{1}^{-1}$$
* $$a^{m}a^{n}=a^{m+n}$$，$${\left(a^m\right)}^n=a^{mn}$$
* $$x,y\in G$$，$$G$$为Abel群，$${\left(xy\right)}^n=x^ny^n$$
* $$\left{ \begin{array}{**lr**} ax=b\ ya=b \end{array} \right.$$在$$G$$中有解，$$G$$满足结合律$$\Longleftrightarrow G$$为一个群

## **子群**

* 定义
  * 子群：$$H$$为$$G$$的一个子集，$$H$$为一个群，记作$$H\leq G$$
  * 平凡子群：$$H=\left{e\right}$$和$$H=G$$
  * 真子群：$$H$$不是平凡子群
* 性质
  * $$H\leq G\Longleftrightarrow\left{ \begin{array}{**lr**} H是满足G下的封闭二元运算\ G的单位元在H内\ \forall a\in H,a^{-1}\in H \end{array} \right.$$
  * $$H\leq G\Longleftrightarrow \forall a,b\in H,ab^{-1}\in H$$
  * $$H\_1,H\_2\leq G\Longrightarrow H\_1\cap H\_2\leq G$$
* 生成
  * $$X$$为$$G$$子集，设$${\left{H\_i\right}}*{i\in I}$$为$$G$$的包含$$X$$的所有子群，则$$\cap*{i\in I}{H\_i}$$为$$G$$的由$$X$$生成的子群，记作$$<X>$$
    * $$X$$的元素为$$<X>$$生成元
    * 若$$G=\<a\_1,\cdots,a\_n>$$，则$$G$$为有限生成的
    * 若$$G=<a>$$，则$$G$$为$$a$$生成的循环群
  * $$G$$为交换群，$$X=\<a\_1,\cdots,a\_t>$$，$$<X>=\left{ \begin{array}{**lr**} \left{a\_{1}^{n\_1}\cdots a\_{t}^{n\_t}\mid a\_i\in X,n\_i\in Z,1\leq i\leq t\right} \&G为乘法群\ \left{n\_1a\_{1}\cdots n\_ta\_{t}\mid a\_i\in X,n\_i\in Z,1\leq i\leq t\right} \&G为加法群 \end{array} \right.$$，

    特别的，$$\forall a\in G,<a>=\left{ \begin{array}{**lr**} \left{a^n\mid n\in Z\right} \&G为乘法群\ \left{na\mid n\in Z\right} \&G为加法群 \end{array} \right.$$

> $$\left(Z\_m,+\right)$$的所有子群
>
> * 对$$n\neq m$$且$$n\mid m$$，$$<n>$$为子群
> * $$<0>=\left{0\right}$$
>
> 乘法群$$Z\_{p}^{\*}$$的所有子群和生成元
>
> * **STEP 1:** $$p-1=q\_1\cdots q\_s$$，模$$p$$原根为$$g$$
> * **STEP 2:**&#x20;
>   * $$<g>$$生成$$p-1$$阶子群
>   * $$\<g^{q\_i}>$$生成$$\frac{p-1}{q\_i}$$阶子群
>   * $$<1>=\left{1\right}$$
>
> $$Z/nZ^\*$$的所有生成元
>
> * **STEP 1:** 模$$n$$原根为$$g$$
> * **STEP 2:** 求所有$$d$$，$$\left(d,\varphi\left(n\right)\right)=1$$
> * **STEP 3:** 生成元为$$g^d$$


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