Finite groups with many p-regular conjugacy classes

Abstract

Let G be a finite group and let p be a prime. In this paper, we study the structure of finite groups with a large number of p-regular conjugacy classes or, equivalently, a large number of irreducible p-modular representations. We prove sharp lower bounds for this number in terms of p and the p′-part of the order of G which ensure that G is p-solvable. A bound for the p-length is obtained which is sharp for odd primes p. We also prove a new best possible criterion for the existence of a normal Sylow p-subgroup in terms of these quantities.