The average element order and the number of conjugacy classes of finite groups

Abstract

Let o(G) be the average order of the elements of G, where G is a finite group. We show that there is no polynomial lower bound for o(G) in terms of o(N ), where N G, even when G is a prime-power order group and N is abelian. This gives a negative answer to a question of A. Jaikin-Zapirain. By the condition p3/c in this theorem, putting c = 1/2 we obtain a negative answer to Question 1.1 for all primes p 7. But in fact a more careful consideration of the parameters involved gives a negative answer for all primes p 5.Corollary 1.3. Question 1.1 has a negative answer for any prime p 5.