# 互換の積

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

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

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

より

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

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

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

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

となる．(置換の積を参照)

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

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

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

$\left(1,2\right)\left(3,4\right)\left(2,3\right)$

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