Ore's conjecture

Statement
The conjecture is that in a finite simple non-abelian group, every element is a commutator.

Stronger conjectures

 * Thompson's conjecture on the existence in simple groups of a conjugacy class whose square is the whole group

Progress towards the conjecture

 * Ore's conjecture holds for alternating groups: This was proved by Ore himself, in the paper where he introduced the conjecture.