Proving A5 has No Normal Subgroups: Conjugacy Classes Approach

[Obraz35]

Homework Statement

I am interested in proving that A₅ has no normal subgroups except itself and {e}.

The Attempt at a Solution

Some proofs that I have seen use centralizers to do this, but since I haven't gone through that yet I think there should be some say to do it without them.

My approach would be to find the conjugacy classes of A₅ and use their orders to show that there cannot be a normal subgroup in A₅ since a normal subgroup is a union of conjugate classes.
But my main problem is how I should go about finding the conjugacy classes.

Thanks for your help.

 

[matt grime]

There is an identical thread to this in this forum. The advice there is: do you know the conjugacy classes of elements in S_5?