On the p-closedness and p-nilpotency of finite groups

摘要

Let acd(G) and acs(G) denote the average irreducible character degree and the average conjugacy class size, respectively, of a finite group G. The object of this paper is to prove that if acd(G) < 2(p + 1)/(p + 3), then G = O-p(G) x Op'(G), and that if acs(G) < 4(p)/(p + 3), then G = O-p(G) x O-p'(G) with O-p(G) abelian, where p is a prime.