论文标题
与有限组的元素顺序的总和有关的图
A graph related to the sum of element orders of a finite group
论文作者
论文摘要
一个有限的组称为$ψ$ -Divisible iff $ψ（h）|ψ（g）$对于有限组$ g $的任何子组$ h $。在这里，$ψ（g）$是$ g $的元素订单的总和。目前，此类组的唯一已知示例是无方顺序的循环。非亚伯利亚$ψ$ - 可分别的群体的存在仍然是一个悬而未决的问题。本文的目的是建立$ψ$可见性属性和图形论之间的联系。因此，对于有限的组$ g $，我们引入了一个简单的无向图，称为$ g $的$ψ$可见度图。我们用$ψ_g$表示。它的顶点是$ g $的非平凡子组，而如果$ h \ subset k $和$ψ（h）|ψ（k）$或$ k \ subset H $和$ n $和$ n $和$ n $和$ n $和$ n $和$ n $和$ψ（k）|ψ（k）|ψ（h）$。我们证明，$ g $是$ψ$ -Divisible Iff $ψ_g$具有通用（主导）顶点。此外，当$ g $是有限的环状群时，我们研究$ψ_g$的各种属性。在本文中，将我们的研究限制在这一特定类别的群体中的选择是为了动机。
A finite group is called $ψ$-divisible iff $ψ(H)|ψ(G)$ for any subgroup $H$ of a finite group $G$. Here, $ψ(G)$ is the sum of element orders of $G$. For now, the only known examples of such groups are the cyclic ones of square-free order. The existence of non-abelian $ψ$-divisible groups still constitutes an open question. The aim of this paper is to make a connection between the $ψ$-divisibility property and graph theory. Hence, for a finite group $G$, we introduce a simple undirected graph called the $ψ$-divisibility graph of $G$. We denote it by $ψ_G$. Its vertices are the non-trivial subgroups of $G$, while two distinct vertices $H$ and $K$ are adjacent iff $H\subset K$ and $ψ(H)|ψ(K)$ or $K\subset H$ and $ψ(K)|ψ(H)$. We prove that $G$ is $ψ$-divisible iff $ψ_G$ has a universal (dominating) vertex. Also, we study various properties of $ψ_G$, when $G$ is a finite cyclic group. The choice of restricting our study to this specific class of groups is motivated in the paper.
