On the average character degree and the average class size in 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 main objective of this paper is to find the lower bounds of acd(G) and acs(G) for a non-r-solvable group G where r is a prime. We show that PSL(2, r), where r >= 5, is a non-r-solvable group with the smallest average character degree and the smallest average class size. (C) 2014 Elsevier Inc. All rights reserved.