Planarity of the intersection graph of subgroups of a finite group

Abstract

For a nontrivial finite group [Formula: see text] different from a cyclic group of prime order, the intersection graph [Formula: see text] of [Formula: see text] is the simple undirected graph whose vertices are the nontrivial proper subgroups of [Formula: see text] and two vertices are joined by an edge if and only if they have a nontrivial intersection. In this paper we characterize all finite groups with planar intersection graphs. It turns out that few solvable groups have planar intersection graphs. Also we classify finite groups whose intersection graphs are bipartite, triangle free and forests.