Intersection graphs of general linear groups

Abstract

Let [Formula: see text] be a field and [Formula: see text] the general linear group of degree [Formula: see text] over [Formula: see text]. The intersection graph [Formula: see text] of [Formula: see text] is a simple undirected graph whose vertex set includes all nontrivial proper subgroups of [Formula: see text]. Two vertices [Formula: see text] and [Formula: see text] of [Formula: see text] are adjacent if [Formula: see text] and [Formula: see text]. In this paper, we show that if [Formula: see text] is a finite field containing at least three elements, then the diameter [Formula: see text] is [Formula: see text] or [Formula: see text]. We also classify [Formula: see text] according to [Formula: see text]. In case [Formula: see text] is infinite, we prove that [Formula: see text] is one-ended of diameter 2 and its unique end is thick.

Low Diameter Algebraic Graphs

“…When n = 1, the requirement for D to be infinite cannot be relieved: Γ (F * 7 = GL 1 (F 7 )) is not a connected graph. In the sequel, we compare Theorem 1 with known results when D = F is a field [2]. A division ring D with center F is weakly locally finite if for every finite subset S of D the division subring F (S) generated by S over F is a finite dimensional vector space over its center [9].…”

“…Here, the general linear group is the set of invertible square matrices of size n. Its subgroups are sets of invertible square matrices closed under multiplication and inversion. When n ≥ 2 and D = F is a field with at least 3 elements, Bien and Viet recently proved that Γ (GL n (F )) has diameter either 2 or 3 [2].…”