Sign of a power of a permutation in $S_n$

Question

I saw the following statement:

no possible $\sigma\in S_4$ so $\sigma^{2}=(1,2,3,4)$ because $\sigma^2$ is even (prove it).

It made me wonder what can we say about sign of $\sigma^k$ permutation in $S_n$? I'm looking for some property to set that $\sigma^k$ is odd or even.

On another note, how to prove formally that $\sigma^2$ is even?

I would try to prove it like this:

• If $\sigma$ is even then $\sigma^2$ is even (not sure because of my main question).
• If $\sigma$ is odd then $\sigma^2$ is even (not sure because of my main question).

So $\sigma^2$ is always even and $(1,2,3,4)$ is odd so its impossible.

Answer 1

Well, the sign of a permutation $\pi$ is $1$ if $\pi$ can be written by a product of an even number of transpositions, and $-1$ otherwise. This is well-defined, since if a permutation can be written by a product of an even (odd) number of transpositions, every other representation of $\pi$ is also a product of an even (odd) number of transpostions.

Now we have ${\rm sgn}(\pi) = (-1)^k$ if $\pi$ can be written as a product of $k$ transpositions.

From here it follows that ${\rm sgn}(\pi\sigma) = {\rm sgn}(\pi)\cdot {\rm sgn}(\sigma)$ and so ${\rm sgn}(\pi^k) = {\rm sgn}(\pi)^k$. This should answer your question.