> For the complete documentation index, see [llms.txt](https://chenyangwang.gitbook.io/mathematical-base-for-information-safety/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://chenyangwang.gitbook.io/mathematical-base-for-information-safety/qun/zhi-huan-qun.md).

# 置换群

## $$n$$**元置换**

> 设$$S=\left{1,\cdots,n\right},\sigma:S\rightarrow S,k\rightarrow \sigma{\left(k\right)=i\_k}$$，表示为$$\sigma=\left( \begin{array}{ccc} 1&2&\cdots\&n-1\&n\ i\_1\&i\_2&\cdots\&i\_{n-1}\&i\_n \end{array} \right)$$

* $$\sigma^{-1}=\left( \begin{array}{ccc} i\_1\&i\_2&\cdots\&i\_{n-1}\&i\_n\ 1&2&\cdots\&n-1\&n \end{array} \right)$$
* 若$$S$$中部分元素$$\left{i\_1,\cdots,i\_k\right}$$满足$$\sigma{\left(i\_1\right)}=i\_2,\sigma{\left(i\_2\right)}=i\_3,\cdots,\sigma{\left(i\_k\right)}=i\_1$$，则称为k-轮变换，简称轮换，记作$$\sigma=\left(i\_1,\cdots,i\_k\right)$$
  * $$k=1$$时为恒等置换
  * $$k=2$$时为对换
  * $$\sigma=\left(i\_1,\cdots,i\_k\right),\tau=\left(j\_1,\cdots,j\_l\right)$$，若$$k+l$$个元素均不相同，则$$\sigma,\tau$$不相交
  * 任意一个置换都可以表示为一些不相交轮换的乘积，且表达式唯一
  * k-轮换可以表示为2-轮换，$$\left(a\_1\cdots,a\_k\right)=\left(a\_1,a\_k\right)\left(a\_1,a\_{k-1}\right)\cdots\left(a\_1,a\_2\right)$$

## **置换群**

$$n$$元置换全体组成的集合$$S\_n$$置换的乘法构成$$n$$元置换群，阶为$$n!$$

设$$G$$为$$n$$元群，则$$G$$同构一个$$n$$元置换群


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