SOME RESULTS ON THE COMPLEMENT OF THE INTERSECTION GRAPH OF SUBGROUPS OF A FINITE GROUP

In this article we consider groups G such that G admits at least one nontrivial subgroup (recall that a subgroup H of G is said to be nontrivial if H /∈ {G, {e}}). Let G be a group. Recall that the intersection graph of subgroups of G, denoted by Γ(G), is an undirected graph whose vertex set is the set of all nontrivial subgroups of G and distinct vertices H, K are joined by an edge in this graph if and only if H ∩ K ̸= {e}. Let G be a finite group. The aim of this article is to investigate the interplay between the group-theoretic properties of a finite group G and the graph-theoretic properties of the complement of Γ(G).