Question: Take two elements of $A_n$, say $a$ and $b$ such that $a$ and $b$ have the same cycle structure. Assume that in the cycle decomposition of $a$ and $b$, two cycles have the same length. Can we necessarily say that $a$ and $b$ are conjugate in $A_n$?
I know that in $S_n$, if two elements have the same cycle structure, then they must be in the same conjugacy class since different cycle structures will give different conjugacy classes where the order of each conjugacy class is $n!$ divided by the least common multiple of the number of elements in each cycle in the cycle decompositions. So, since every element in $A_n$ can be written in an even number of transpositions, I was hoping to be able to use this to help show that, but I can't seem to get it to work. Any help is greatly appreciated!