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.

Let K be a field and let V be a vector space of dimension 2m over K. Let ∧V denote the exterior algebra of V and ∧kV its kth exterior power for 0[les ]k[les ]2m. Let f be a non-degenerate alternating bilinear form defined on V×V. The symplectic group Sp2m(K) is the group of all isometries of f and it acts as a group of vector space automorphisms on ∧kV. In the case that K is algebraically closed and 1[les ]k[les ]m, it is known that ∧kV contains a composition factor corresponding to the fundamental weight ωk of a root system of type Cm. We shall refer to the irreducible module for Sp2m(K) given by this composition factor as a fundamental module.