On a divisibility property involving the sum of element orders

A finite group $G$ is called $\psi$-divisible if $\psi (H)\mathrm{\mid }\psi (G)$ for any subgroup $H$ of $G$, where $\psi (H)$ and $\psi (G)$ are the sum of element orders of $H$ and $G$, respectively. In this paper, we extend a result provided in [10], by classifying the finite groups whose all subgroups are $\psi$-divisible. Since the existence of $\psi$-divisible groups is related to the class of square-free order groups, we also study the sum of element orders and the $\psi$-divisibility property of ZM-groups. In the end, we introduce the concept of $\psi$-normal divisible group, i.e. a group for which the $\psi$-divisibility property is satisfied by all its normal subgroups. Using simple and quasisimple groups, we are able to construct infinitely many $\psi$-normal divisible groups which are neither simple nor nilpotent.