Intersection graphs of quasinormal subgroups of general skew linear groups

Year 2024, Volume: 53 Issue: 2, 392 - 404, 23.04.2024

Le Qui Danh

https://doi.org/10.15672/hujms.1249433

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

Supporting Institution

Vietnam National University HoChiMinh City (VNUHCM)

Project Number

T2022-18-03

Thanks

This research is funded by Vietnam National University HoChiMinh City (VNUHCM) under grant number T2022-18-03.

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