Commutators in Finite Simple Groups of Lie Type

Using properties of the Steinberg character, we obtain a congruence modulo p for the number of ways in which a p ‐regular element may be expressed as a commutator in a finite simple group G of Lie type of characteristic p . This congruence shows that such an element is a commutator in G . We also show that if K and L are conjugacy classes of G consisting of elements whose centralizers have order relatively prime to p , then any p ‐regular element of G is expressible as the product of an element of K and an element of L . 1991 Mathematics Subject Classification 20C20, 20D08, 20G40.