= Permutation conjugate to its square
{title2=$\sigma\sim_{S_n}\sigma^2\ \Longleftrightarrow\ \operatorname{ord}(\sigma)\text{ is odd}$}
Squaring an odd-length <permutation cycle> retains its length, whereas an even-length cycle splits into two cycles of half its length. Thus if any even cycle is present, the number of cycles strictly increases, so the <cycle type> changes. Since <conjugate permutations> in a <symmetric group> have the same cycle type, a permutation is conjugate to its square precisely when all its cycle lengths are odd, equivalently when its <order of a group element> is odd.
Back to article page
