Some density results involving the average order of a finite group

Abstract

Let $o(G)$ be the average of the element orders of a finite group G. A research topic concerning this quantity is understanding the relation between $o(G)$ and $o(H)$, where H is a subgroup of G. Let $\mathcal{N}$ be the class of finite nilpotent groups and let $L(G)$ be the subgroup lattice of G. In this paper, we show that the set $\left\{\frac{o(G)}{o(H)}\right|G\in \mathcal{N},H\in L(G)\}$ is dense in $[0,\infty )$. Other density results are outlined throughout the paper.

KEYWORDS:

• Density of a set
• element orders
• nilpotent groups
• p-groups

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 20D15
• Secondary: 20D60
• 40A05

Acknowledgments

The author is grateful to the reviewers for their remarks which improve the previous version of the paper.

Additional information

Funding

This work was supported by a grant of the “Alexandru Ioan Cuza” University of Iasi, within the Research Grants program, Grant UAIC, code GI-UAIC-2021-01.