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 $G$.
This problem is difficulty, I thinking about $A_n$ in $S_n$ ($n>4$), but there is not some ideas. I hope someone give advice, or some old results about this studying.