Splitting criterion for conjugacy classes in the alternating group

Statement

Suppose $n$ is a natural number. Let $S_{n}$ be the symmetric group on $n$ letters, and $A_{n}$ be the alternating group: the subgroup of even permutations. Then, given a conjugacy class in $S_{n}$ of even permutations (i.e., a conjugacy class that lies completely inside $A_{n}$ ), the following two possibilities hold:

This conjugacy class is equal to a single conjugacy class in $A_{n}$ .
This conjugacy class splits into two conjugacy classes in $A_{n}$ .

The criterion that determines the fate of a conjugacy class (i.e., whether or not it splits) is termed the splitting criterion, and this criterion can be formulated both in the language of centralizers and the language of cycle decompositions.

In terms of centralizers

The conjugacy class of an element $g \in A_{n}$ :

splits if the centralizer $C_{S_{n}} (g)$ of $g$ in $S_{n}$ is contained in $A_{n}$ , i.e., if any permutation that commutes with $g$ is even; and
does not split if the centralizer $C_{S_{n}} (g)$ of $g$ in $S_{n}$ is not contained in $A_{n}$ , i.e., if there exists an odd permutation that commutes with $g$ .

In terms of cycle decompositions

The conjugacy class of an element $g \in A_{n}$ :

splits if the cycle decomposition of $g$ comprises cycles of distinct odd length. Note that the fixed points are here treated as cycles of length $1$ , so it cannot have more than one fixed point; and
does not split if the cycle decomposition of $g$ contains an even cycle or contains two cycles of the same length.