A note on powers in simple groups

In [ 7 ], the second author proved that there is an integer k such that every element of a finite non-abelian simple group S is a product of k commutators in S . The motivation for proving this result came from a model-theoretic question about simple groups. The proof depended on the classification of the finite simple groups, a theorem of Malle, Saxl and Weigel [ 5 ] which shows that in many finite simple classical groups S there is a real conjugacy class R such that S = R 3 ∪{1}, and an ultraproduct argument. Here we shall use a similar combination of ideas to prove the following result.