The Clique Number of the Intersection Graph of a Finite Group

For a nontrivial finite group G, the intersection graph (G) of G is the simple undirected graph whose vertices are the nontrivial proper subgroups of G and two vertices are joined by an edge if and only if they have a nontrivial intersection. In a finite simple graph , the clique number of is denoted by ω( ). In this paper we show that if G is a finite group with ω( (G)) 13, then G is solvable. As an application, we characterize all non-solvable groups G with ω( (G)) = 13. Moreover, we determine all finite groups G with ω( (G)) ∈ {2, 3, 4}.