# 執位

## 例子

${\displaystyle \phi (1)=1,\phi (2)=4,\phi (3)=2,\phi (4)=3}$

${\displaystyle \phi ={\begin{bmatrix}1&2&3&4\\\phi (1)&\phi (2)&\phi (3)&\phi (4)\end{bmatrix}}={\begin{bmatrix}1&2&3&4\\1&4&2&3\end{bmatrix}}}$

${\displaystyle \phi ={\begin{bmatrix}1&2&3&4&5\\2&4&3&5&1\\\end{bmatrix}};\sigma ={\begin{bmatrix}1&2&3&4&5\\5&4&1&2&3\\\end{bmatrix}}}$

${\displaystyle \sigma \phi ={\begin{bmatrix}1&2&3&4&5\\2&4&3&5&1\\\end{bmatrix}}{\begin{bmatrix}1&2&3&4&5\\5&4&1&2&3\\\end{bmatrix}}={\begin{bmatrix}1&2&3&4&5\\1&5&2&4&3\\\end{bmatrix}}}$

${\displaystyle \sigma \phi }$ 嘅意思係${\displaystyle \sigma \circ \phi }$ ，先做${\displaystyle \phi }$ 再做${\displaystyle \sigma }$ 。 所以，${\displaystyle \sigma \phi (1)=\sigma (\phi (1))=\sigma (5)=1}$

### 對稱群一

${\displaystyle e={\begin{bmatrix}1&2&3\\1&2&3\\\end{bmatrix}}}$ ${\displaystyle a={\begin{bmatrix}1&2&3\\2&3&1\\\end{bmatrix}}}$ ${\displaystyle a^{2}={\begin{bmatrix}1&2&3\\3&1&2\\\end{bmatrix}}}$

${\displaystyle b={\begin{bmatrix}1&2&3\\1&3&2\\\end{bmatrix}}}$ ${\displaystyle ab={\begin{bmatrix}1&2&3\\2&1&3\\\end{bmatrix}}}$ ${\displaystyle a^{2}b={\begin{bmatrix}1&2&3\\3&2&1\\\end{bmatrix}}}$

${\displaystyle ab={\begin{bmatrix}1&2&3\\2&3&1\\\end{bmatrix}}{\begin{bmatrix}1&2&3\\1&3&2\\\end{bmatrix}}={\begin{bmatrix}1&2&3\\2&1&3\\\end{bmatrix}}}$

${\displaystyle ba={\begin{bmatrix}1&2&3\\1&3&2\\\end{bmatrix}}{\begin{bmatrix}1&2&3\\2&3&1\\\end{bmatrix}}={\begin{bmatrix}1&2&3\\1&3&2\\\end{bmatrix}}}$

### 對稱群二

${\displaystyle S_{n}}$ 嘅嘢係咁嘅樣${\displaystyle \sigma ={\begin{bmatrix}1&2&\cdots &n\\\sigma (1)&\sigma (2)&\cdots &\sigma (n)\end{bmatrix}}}$

### 正方形旋轉反射群

${\displaystyle D_{4}}$ ${\displaystyle S_{4}}$ 嘅子群。