Intersection graphs of quasinormal subgroups of general skew linear groups

The intersection graph of quasinormal subgroups of a group G , denoted by Gamma(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 boolean AND K is nontrivial. In this paper, we show that when G is an arbitrary nonsimple group, the diameter of Gamma(q)(G) is in {0,1,2,infinity}. Besides, all general skew linear groups GL(n)(D) over a division ring D can be classified depending on the diameter of Gamma(q)(GL(n)(D)).