On Groups with the Second Largest Value of the Sum of Element Orders

Abstract

For a finite group G, let ψ(G) denote the sum of element orders of G. It is known that the maximum value of ψ on the set of groups of order n, where n is a positive integer, will occur at the cyclic group ℤ _n . In this paper, we investigate groups with the second largest value of ψ on the set of groups of the same order.

Key Words:

• Element orders
• Finite groups
• Metacyclic groups

2010 Mathematics Subject Classification:

• 20D45

ACKNOWLEDGMENTS

The work in this paper was mostly undertaken while the first author was visiting Auckland University. The first author thanks Professor Marston Conder for his hospitality reception and for giving some examples by Magama. We also thank the referee's comments and Professor Yun Fan's discussion.

Dedicated to Professor Wujie Shi on the occasion of his 70th birthday.

Notes

Communicated by A. Turull.