Finite p-solvable groups with three p-regular conjugacy class sizes

Abstract

Let G be a finite p-solvable group. We describe the structure of the p-complements of G when the set of p-regular conjugacy classes has exactly three class sizes. For instance, when the set of p-regular class sizes of G is {1, pa , pam} or {1, m, pam} with (m, p) = 1, then we show that m = qb for some prime q and the structure of the p-complements of G is determined. Copyright © Edinburgh Mathematical Society 2013.