On the number of irreducible modular representations of a finite group

Abstract

1. Let K be a field of characteristic p, and let G be a finite group. Denote by n the L.C.M. of the orders of the p-regular elements of G, and let 6 be a primitive nth root of 1 over K. Each K-automorphism of K(a) is determined by a map a,->5a for some integer t, taken modulo n. The multiplicative group T of all such exponents t (mod n) is isomorphic to the Galois group of K(6) over K. Two p-regular elements a, b E G are called K-conjugate if bt = x-lax for some xEG and some tE T. This defines an equivalence relation, relative to which the p-regular elements of G are partitioned into pregular K-conjugacy classes. The following is due to Berman [2].