Show that $\sigma^2$ is a $p$-cycle when $p$ is odd, and $\sigma^2$ is not a cycle when $p$ is even with $p \neq 2$

Question

Let $\sigma$ be a $p$-cycle. Show that:

• $\sigma^2$ is a $p$-cycle when $p$ is odd, and
• $\sigma^2$ is not a cycle when $p$ is even with $p \neq 2$

I'm learning about permutations and cycles, and have been working on this problem, but haven't been able to solve it.

Answer 1

Let $\sigma=a_1a_2\dots a_p$. $\sigma^2$ can be read off from $\sigma$ easily; $a_i$ maps to $a_{i+2\bmod p}$. Thus if $p=2k+1$, $\sigma^2=a_1\dots a_{2k+1}a_2\dots a_{2k}$, which is again a $p$-cycle. If $p$ is even then $\sigma^2$ is two disjoint cycles, consisting of the even- and odd-indexed $a_i$ in each. (The second statement is true even when $p=2$, for in that case $\sigma^2$ is the identity permutation and not a single cycle.)