Intersection graphs of general linear groups

Let F be a field and GL(n)(F) the general linear group of degree n > 1 over F. The intersection graph Gamma(GL(n)(F)) of GL(n)(F) is a simple undirected graph whose vertex set includes all nontrivial proper subgroups of GL(n)(F). Two vertices A and B of Gamma(GL(n)(F)) are adjacent if A not equal B and A boolean AND B not equal{In}. In this paper, we show that if F is a finite field containing at least three elements, then the diameter delta(Gamma(GL(n)(F))) is 2 or 3. We also classify GL(n)(F) according to delta(Gamma(GL(n)(F))). In case F is infinite, we prove that Gamma(GL(n)(F)) is one-ended of diameter 2 and its unique end is thick.