Structure of intersection graphs

 Let G be a finite group and let N be a fixed normal subgroup of G.  In this paper, a new kind of graph on G, namely the intersection graph is defined and studied. We use  to denote this graph, with its vertices are all normal subgroups of G and two distinct vertices are adjacent if their intersection in N. We show some properties of this graph. For instance, the intersection graph is a simple connected with diameter at most two. Furthermore we give the graph structure of  for some finite groups such as the symmetric, dihedral, special linear group, quaternion and cyclic groups.