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

2020/09/17 by E. I. Khukhro, Khukhro, E. I., Alexander Moretó +3
Computer Science · Engineering · Mathematics · #20C15 #20D15 #20E45 #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2009.08226

openalex publication_date 2020/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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\trianglelefteq 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.

Related

• On the average order of a finite group
• Finite groups have even more conjugacy classes
• A lower bound for the number of conjugacy classes of finite groups
• Characterizing some finite groups by the average order
• An affirmative answer to the Jainkin-Zapirain's question