On the average character degree and the average class size in finite groups

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.