On the diameter of intersection graphs of finite groups

The intersection graph G of a finite group G is a simple graph with vertices the non-trivial proper subgroups of G, and an edge between two vertices if their corresponding subgroups intersect non-trivially. These graphs were introduced by Cs\'ak\'any and Poll\'ak in 1969. In this paper we answer two long-standing open questions posed by Cs\'ak\'any and Poll\'ak concerning the diameter of intersection graphs. We prove some necessary conditions for a non-simple group to have an intersection graph of diameter 4. We also construct the first examples of non-simple groups and alternating groups whose intersection graphs have diameter 4.