Øystein Ore proved in 1951 that for $n\gt4$ we have that all elements of $A_n$ are commutators. He subsequently conjectured that this is true for all finite non-abelian simple groups.
In this connection, not long ago I asked if every element of a perfect group is a commutator. It was answered in the negative. I should have caught that $A_5*A_5$ is a counterexample. The smallest such was found to be of order 960.
Ore's conjecture/theorem was proved in 2008, and relies on the classification theorem.
- See M. W. Libeck, E. A. O'Brien, Aner Shalev, Pham Huu Tiep, The Ore conjecture, Journal of the European Mathematical Society, vol. 12, issue 4, 2010. pp. 939-1008. http://doi.org/10.4171/JEMS/220
Interestingly Ore himself was mainly concerned with graph theory.
