On the structure of finite groups determined by the arithmetic and geometric means of element orders

2022/12/28 by Valentina Grazian, Grazian, Valentina, Carmine Monetta +3 · 2 citations
Engineering · Mathematics · #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Rings, Modules, and Algebras #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2212.13770

openalex publication_date 2022/12/28 · openalex created_date 2023/01/06 · openalex updated_date 2026/07/28

Abstract

In this paper we consider two functions related to the arithmetic and geometric means of element orders of a finite group, showing that certain lower bounds on such functions strongly affect the group structure. In particular, for every prime p, we prove a sufficient condition for a finite group to be p-nilpotent, that is, a group whose elements of p'-order form a normal subgroup. Moreover, we characterize finite cyclic groups with prescribed number of prime divisors.

Cited by

• The influence of the prime omega function on the product of element orders in finite groups
• Recognition by the set of exponents in the prime factorization of the product of element orders

Related

• A criterion for nilpotency of a finite group by the sum of element orders
• A Sufficient Condition for Nilpotency in a Finite Group
• A p-nilpotency criterion for finite groups
• A result on certain sums of element orders in finite groups
• Detecting structural properties of finite groups by the sum of element orders