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On a Divisibility Property Involving the Sum of Element Orders
Bulletin of the Malaysian Mathematical Sciences Society ( IF 1.0 ) Pub Date : 2020-08-08 , DOI: 10.1007/s40840-020-00987-8
Mihai-Silviu Lazorec

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.



中文翻译:

关于涉及元素序数总和的可除性

有限群G ^称为\(\ PSI \) -divisible如果\ |(\ PSI(H)\ PSI(G)\)为任何亚组ħģ,其中\(\ PSI(H)\)\( \ psi(G)\)分别是HG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。

更新日期:2020-08-09
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