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arXiv · 2212.13770

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

Valentina Grazian·Carmine Monetta·Marialaura Noce
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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.

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Valentina Grazian, Carmine Monetta, Marialaura Noce. 2022-12-28. On the structure of finite groups determined by the arithmetic and geometric means of element orders. https://arxiv.org/abs/2212.13770

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