I am a new user and I don’t know how to use Latex, so I apologize for my text first. I have two questions.
1: I know that the conjugate of a cycle in $S_n$ will not change its cycle type. And I know the converse is also true, i.e. all the cycle with same cycle type consists of a conjugacy class. But I don’t know how to prove this converse. Is there anyone can help me with this?
As above, we can write down all the conjugacy classes of $S_n$ easily since we just need to find all the types of cycle, which exactly corresponds to all the partitions of
1, 2, ..., n.
The question is that how we find all the conjugacy classes of $A_n$? The method in dealing with $S_n$ doesn’t work now, since there always are two different cycles with the same cycle type, but they are conjugated by a odd cycle. I think about this for a time, but I can’t find a way to formulate it.
So is there someone can help me with this?
Appreciate so much :)
Another detailed proof see: Splitting of conjugacy class in alternating group , as Ethan Bolker quotes.