What I want to prove is... A conjugacy class in $S_n$ splits into two distinct conjugacy classes under the action of $A_n$ if and only if its cycle type consists of distinct odd integers. Otherwise, it remains a single conjugacy class in $A_n$.
To prove this statement I proved...
• A conjugacy class in $S_n$ consisting solely of even permutations (i.e. that is contained in $A_n$ ) either is a single conjugacy class or is the disjoint union of two equal-sized conjugacy classes when considered under the action of $A_n$.
• If $\sigma \in A_n$, then the elements of the conjugacy class of $\sigma \in S_n$ (which is just all elements of the same cycle type as $\sigma$ ) are conjugate in $A_n$ if and only if $\sigma$ commutes with some odd permutation.
• $\sigma \in S_n$ does not commute with an odd permutation if and only if the cycle type of $\sigma$ consists of distinct odd integers.
I proved first two lemma and only if part of the third lemma. But I got problem considering if part of the third lemma. The cycle type of $(1 2 3)$ in $A_{n}$ is $(3)$. But it commutes with $(4 5)$ since $(1 2 3)$ and $(4 5)$ are disjoint. I'm not sure that I'm confusing something or the author of https://planetmath.org/conjugacyinan is saying something wrong. I hope somebody help me.
