Intersection graphs of quasinormal subgroups of general skew linear groups

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