互換の積

恒等置換 ${\displaystyle \left(\begin{array}{llll}1\hfill & 2\hfill & 3\hfill & 4\hfill \\ 1\hfill & 2\hfill & 3\hfill & 4\hfill \end{array}\right)}$ に互換 ${\displaystyle \left(1,2\right)}$ ，${\displaystyle \left(3,4\right)}$ ，${\displaystyle \left(2,3\right)}$ を順にかけると

${\displaystyle \left(1,2\right)=\left(\begin{array}{llll}1\hfill & 2\hfill & 3\hfill & 4\hfill \\ 2\hfill & 1\hfill & 3\hfill & 4\hfill \end{array}\right)}$

${\displaystyle \left(3,4\right)=\left(\begin{array}{llll}1\hfill & 2\hfill & 3\hfill & 4\hfill \\ 1\hfill & 2\hfill & 4\hfill & 3\hfill \end{array}\right)}$

${\displaystyle \left(2,3\right)=\left(\begin{array}{llll}1\hfill & 2\hfill & 3\hfill & 4\hfill \\ 1\hfill & 3\hfill & 2\hfill & 4\hfill \end{array}\right)}$

より

${\displaystyle \left(\begin{array}{llll}1\hfill & 2\hfill & 3\hfill & 4\hfill \\ 1\hfill & 2\hfill & 3\hfill & 4\hfill \end{array}\right)\left(1,2\right)\left(3,4\right)\left(2,3\right)}$

${\displaystyle =\left(\begin{array}{llll}1\hfill & 2\hfill & 3\hfill & 4\hfill \\ 3\hfill & 1\hfill & 4\hfill & 2\hfill \end{array}\right)}$

となる．(置換の積を参照)

${\displaystyle \left(\begin{array}{llll}1\hfill & 2\hfill & 3\hfill & 4\hfill \\ 3\hfill & 1\hfill & 4\hfill & 2\hfill \end{array}\right)}$${\displaystyle =\left(\begin{array}{llll}1\hfill & 2\hfill & 3\hfill & 4\hfill \\ 1\hfill & 2\hfill & 3\hfill & 4\hfill \end{array}\right)\left(1,2\right)\left(3,4\right)\left(2,3\right)}$

${\displaystyle =I\left(1,2\right)\left(3,4\right)\left(2,3\right)}$

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

恒等置換Iを省略して　

${\displaystyle \left(1,2\right)\left(3,4\right)\left(2,3\right)}$

を互換の積という．

置換　 ${\displaystyle \left(\begin{array}{llll}1\hfill & 2\hfill & 3\hfill & 4\hfill \\ 3\hfill & 1\hfill & 4\hfill & 2\hfill \end{array}\right)}$ 　は互換の積　 ${\displaystyle \left(1,2\right)\left(3,4\right)\left(2,3\right)}$ 　で表せされる．

 

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最終更新日： 2025年1月17日