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

TL;DR

This paper investigates how bounds on functions related to the arithmetic and geometric means of element orders influence the structure of finite groups, providing conditions for p-nilpotency and characterizations of cyclic groups.

Contribution

It introduces new conditions based on element order means that determine p-nilpotency and characterizes cyclic groups with specific prime divisor counts.

Findings

Lower bounds on mean functions imply p-nilpotency

Characterization of cyclic groups with given prime divisors

Conditions for group structure based on element order means

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.