TITLE: On J.G.Thompson's conjecture about conjugacy classes of finite simple groups
QUESTION [5 upvotes]: This conjecture in "Unsolved problems in group theory" No.18: 9.24:
Conjecture: every finite simple non-abelian group $G$ can be represented in the form $G=CC$, where $C$ is some conjugacy class of $G$.
I want to prove this conjecture is right for simple group $A_n$  in $S_n$ ($n>4$), but I have no idea how to deal with it. I hope someone could give me some advice, or some old results about this.

REPLY [5 votes]: This was proved in this paper:
MR0183763 (32 #1241) Reviewed 
Xu Cheng-hao
The commutators of the alternating group. 
Sci. Sinica 14 1965 339–342. 
20.20 

My institution does not have an electronic copy, so there you are on your own.