Exercise 3.5.2 ($\sigma^2$ is an even permutation)

Question

Prove that $\sigma^2$ is an even permutation for every permutation $\sigma$.

Richard Ganaye · Accepted Answer

\begin{proof} 
The permutation $\sigma$ may be written as a product of transpositions $	au_1,	au_2,\cdots, 	au_k$:
$$\sigma = 	au_1	au_2\cdots 	au_k.$$
Then 
$$\sigma^2 = 	au_1	au_2\cdots 	au_k	au_1	au_2\cdots 	au_k$$
is the product of $2k$ transpositions, so $\sigma^2$ is an even permutation.

\end{proof}

Exercise 3.5.2 ($\sigma^2$ is an even permutation)

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Exercise 3.5.2 ($\sigma^2$ is an even permutation)

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