论文标题
在涉及元素订单之和的划分属性上
On a divisibility property involving the sum of element orders
论文作者
论文摘要
如果任何子组$ h $的$ h $ of $ g $，则有限的组$ g $称为$ψ$ - 可见，其中$ψ（h）$和$ψ（g）$分别是$ h $和$ g $的元素订单的总和。在本文中，我们通过对所有子组均为$ψ$ - 可见的有限组进行分类，扩展了[10]中提供的结果。由于存在$ψ$ - 可见组与无方订单组的类别有关，因此我们还研究元素订单的总和和$ψ$ - 可见性的属性。最后，我们介绍了$ψ$ - 正常可划分组的概念，即，其所有正常子组都满足了$ψ$可见性属性的组。使用简单的和准的组，我们能够构建既不简单也不是nilpotent的无限$ψ$正常的可分割组。
A finite group $G$ is called $ψ$-divisible if $ψ(H)|ψ(G)$ for any subgroup $H$ of $G$, where $ψ(H)$ and $ψ(G)$ 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 $ψ$-divisible. Since the existence of $ψ$-divisible groups is related to the class of square-free order groups, we also study the sum of element orders and the $ψ$-divisibility property of ZM-groups. In the end, we introduce the concept of $ψ$-normal divisible group, i.e. a group for which the $ψ$-divisibility property is satisfied by all its normal subgroups. Using simple and quasisimple groups, we are able to construct infinitely many $ψ$-normal divisible groups which are neither simple nor nilpotent.
