During reading, I have encountered this, in several places:
The following are equivalent for a permutation $\sigma \in A_n$:
1) the $S_n$-conjugacy class of $\sigma$ splits into two $A_n$-classes
2) there is no odd permutation which commutes with $\sigma$
3) $\sigma$ has no cycles of even length, and all its cycles have distinct lengths
I don't understand the meaning of the first bullet, what is the meaning of Conjugacy class splitting in two? what does it look like?