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.

On the average order of a finite group

“…Recently, E.I. Khukhro, A. Moretó and M. Zarrin proved the following result (see Theorem 1.2 of [11]): Theorem 1.1. Let c > 0 be a real number and let p ≥ 3 c be a prime.…”

Another criterion for supersolvability of finite groups

“…Recently, E.I. Khukhro, A. Moretó and M. Zarrin proved the following result (see Theorem 1.2 of [11]):…”

“…In the last years there has been a growing interest in studying the properties of these functions and their relations with the structure of G (see for example [1]- [4], [6]- [8], [10]- [11], [14] and [18]- [19]).…”

“…Then there exists a finite p-group G with a normal abelian subgroup N such that o(G) < o(N) c . Note that Theorem A provides a negative answer to Jaikin-Zapirain's question even if we replace the exponent 1 2 with any positive real number c. In the same paper [11], the authors posed the following conjecture:…”

Another criterion for solvability of finite groups

“…Recently many authors studied the function ψ(G) and, more generally, properties of finite groups determined by their element orders (see for example [1]- [5], [9], [12]- [17], [20], [22]- [29], [34], [36] and [38]).…”