关于一个涉及元素阶和的整除性质
On a divisibility property involving the sum of element orders
摘要：
一个有限群$G$称为$ψ$ -可分群，如果对于$G$的任何子群$H$，有$ψ(H)|ψ(G)$，其中$ψ(H)$和$ψ(G)$分别是$H$和$G$的元素阶之和。本文推广了文[10]中的一个结果，对子群均为$ψ$ -可分的有限群进行了分类.由于$ψ$ -可除群的存在性与无平方序群类有关，我们还研究了ZM-群的元素阶和和及$ψ$ -可除性.最后，我们引入了$ψ$ -正规可除群的概念，即所有正规子群都满足$ψ$ -可除性的群。利用单群和拟单群，我们可以构造出无穷多个既不是单群也不是幂零群的$ψ$ -正规可除群。
英文摘要：
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.
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