I know that in $S_n$, two cycles are conjugate if and only if they have the same cycle structure. This isn't true of $A_n$ though because apparently $(123)$ and $(213)$ aren't conjugate in $A_3$.
My question is firstly, how can we prove that $(123)$ and $(213)$ aren't conjugate in $A_3$, and secondly is there some way of generalising that for arbitrary $n$?
By that I mean for any $n$, can we find two permutations of the same cycle type that aren't conjugate in $A_n$?
It turns out that the conjugacy class of an even permutation $g \in S_n$, $n > 1$, decomposes into smaller conjugacy classes in $A_n$ (always into two classes) iff the cycle decomposition of $g$ is into odd cycles of distinct length.
So, if $n > 2$ is odd, then the $n$-cycles split into two conjugacy classes. If $n > 2$ is even, then the $(n - 1)$-cycles split into two classes.
It follows from the above characterization that for $2 < n < 8$ only one cycle type splits, but for $n = 8$ two do (those corresponding to cycle types $(1, 7)$ and $(3, 5)$).