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 $\lbrace \frac{o(G)}{o(H)} \ | \ G\in\mathcal{N}, H\in L(G)\rbrace$ is dense in $[0, \infty)$. Other density results are outlined throughout the paper.