# 置换的奇偶性

${\displaystyle \operatorname {sgn}(\sigma )=(-1)^{N(\sigma )}}$

${\displaystyle \operatorname {sgn}(\sigma )=(-1)^{m}}$

## 例子

${\displaystyle \sigma =(23)(12)(24)(35)(45),\;}$

## 性质

• 两个偶置换的复合是偶的
• 两个奇置换的复合是偶的
• 一个奇置换与偶置换的复合是奇的

• 任何偶置换的逆是偶的
• 任何奇置换的逆是奇的

${\displaystyle \operatorname {sgn} :S_{n}\to \{-1,1\}}$

(a b c d e) = (a e) (b e) (c e) (d e)

## 两个定义的等价性

### 证明一

σ = T'1 T'2 ... T'k'
σ = Q'1 Q'2 ... Q'm'

(2 5) = (2 3)(3 4)(4 5)(4 3)(3 2)

σ = T1 T2 ... Tk
σ = Q1 Q2 ... Qm

### 证明二

${\displaystyle P(x_{1},\ldots ,x_{n})=\prod _{i

${\displaystyle P(x_{1},x_{2},x_{3})=(x_{1}-x_{2})(x_{2}-x_{3})(x_{1}-x_{3}).\;}$

${\displaystyle \operatorname {sgn} (\sigma )={\frac {P(x_{\sigma (1)},\ldots ,x_{\sigma (n)})}{P(x_{1},\ldots ,x_{n})}}}$

${\displaystyle \operatorname {sgn} (\sigma \tau )={\frac {P(x_{\sigma (\tau (1))},\ldots ,x_{\sigma (\tau (n))})}{P(x_{1},\ldots ,x_{n})}}}$
${\displaystyle ={\frac {P(x_{\sigma (1)},\ldots ,x_{\sigma (n)})}{P(x_{1},\ldots ,x_{n})}}\cdot {\frac {P(x_{\sigma (\tau (1))},\ldots ,x_{\sigma (\tau (n))})}{P(x_{\sigma (1)},\ldots ,x_{\sigma (n)})}}}$
${\displaystyle =\operatorname {sgn} (\sigma )\cdot \operatorname {sgn} (\tau )}$

### 证明三

• ${\displaystyle \tau _{i}^{2}=1}$ 对所有i
• ${\displaystyle \tau _{i}\tau _{i+1}\tau _{i}=\tau _{i+1}\tau _{i}\tau _{i+1}}$  对所有i < n − 1，
• ${\displaystyle \tau _{i}\tau _{j}=\tau _{j}\tau _{i}}$  如果 |i − j| ≥ 2。