The Clique Number of the Intersection Graph of a Finite Group.

For a nontrivial finite group G, the intersection graph Γ (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 Γ , the clique number of Γ is denoted by ω (Γ) . In this paper we show that if G is a finite group with ω (Γ (G)) < 13 , then G is solvable. As an application, we characterize all non-solvable groups G with ω (Γ (G)) = 13 . Moreover, we determine all finite groups G with ω (Γ (G)) ∈ { 2 , 3 , 4 } . [ABSTRACT FROM AUTHOR]