Nilpotency of p-complements and p-regular conjugacy class sizes

Abstract

Let G be a finite p-solvable group. We prove that if the set of conjugacy class sizes of all p ′ -elements of G is { 1 , m , p a , m p a } , where m is a positive integer not divisible by p, then the p-complements of G are nilpotent and m is a prime power. This result partially extends a theorem for ordinary classes which asserts that if the set of conjugacy class sizes of a finite group G is exactly { 1 , m , n , m n } and ( m , n ) = 1 , then G is nilpotent.