Intersection graphs of quasinormal subgroups of general skew linear groups

Authors : Le Qui Danh

Pages : 392-404

Doi:10.15672/hujms.1249433

View : 157 | Download : 177

Publication Date : 2024-04-23

Article Type : Research Paper

Abstract :The intersection graph of quasinormal subgroups of a group $G$, denoted by $\\Gamma_{\\mathrm{q}}(G)$, is a graph defined as follows: the vertex set consists of all nontrivial, proper quasinormal subgroups of $G$, and two distinct vertices $H$ and $K$ are adjacent if $H\\cap K$ is nontrivial. In this paper, we show that when $G$ is an arbitrary nonsimple group, the diameter of $\\Gamma_{\\mathrm{q}}(G)$ is in $\\{0,1,2,\\infty\\}$. Besides, all general skew linear groups $\\mathrm{GL}_n(D)$ over a division ring $D$ can be classified depending on the diameter of $\\Gamma_{\\mathrm{q}}(\\mathrm{GL}_n(D))$.
Keywords : division ring, general skew linear group, intersection graph, quasinormal subgroup, permutable subgroup