A finite group G is called \(\psi \)-divisible if \(\psi (H)|\psi (G)\) 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 classify the finite groups whose subgroups are all \(\psi \)-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.

有限群G ^称为\（\ PSI \） -divisible如果\ |（\ PSI（H）\ PSI（G）\）为任何亚组ħ的ģ，其中\（\ PSI（H）\）和\（ \ psi（G）\）分别是H和G的元素顺序之和。在本文中，我们对所有子组都是\（\ psi \）可分割的有限组进行分类。由于\（\ psi \）可整除组的存在与无平方阶数组的类别有关，因此，我们还研究了元素阶数之和和ZM-组的\（\ psi \）-可除性。最后，我们介绍了\（\ psi \） -正常可分割组，即其所有正常子组都满足\（\ psi \）- divisibility属性的组。使用简单和拟简单的组，我们能够构造无数个既不是简单也不是幂等的\（\ psi \） -正态可分组。