THE INCLUSION GRAPH OF SUBGROUPS OF A GROUP

Let G be a group. We assign a graph I(G) to G whose vertex set is all proper non-trivial subgroups of G and two vertices are adjacent if and only if they are comparable. We study those groups whose inclusion graph of subgroups are complete, null, connected, star and triangle-free graph. We prove that if I(G) is a complete graph, then G is isomorphic to Z p n or Z p ∞ . It is shown that if G is an abelian group and I(G) is triangle-free, then j G j 2 f p 2 , p 3 , pq, p 2 q, pqr g , where p, q and r are distinct primes. Moreover, we prove that if G is a finite abelian group and I(G) is a tree, then G is isomorphic to one of the groups Z p 2 , Z p 3 , Z p 2 q for some distinct primes p and q. Also we prove that if G is an infinite solvable group with trivial center, then I(G) is connected. Finally, we show that a finite group G is isomorphic to one of the groups Z p 2 , Z p 3 or Q 8 , when I(G) is a star graph.