Finite abelian groups with isomorphic inclusion graphs of cyclic subgroups

Let G be a finite group. The directed inclusion graph of cyclic subgroups of G, →Ic(G), is the digraph with vertices of all cyclic subgroups of G, and for two distinct cyclic subgroups a and b , there is an arc from a to b if and only if b ⊂ a . The (undirected ) inclusion graph of cyclic subgroups of G, Ic(G), is the underlying graph of →Ic(G), that is, the vertex set is the set of all cyclic subgroups of G and two distinct cyclic subgroups ai and b are adjacent if and only if a ⊂ b or b ⊂ a . In this paper, we first show that, if G and H are finite groups such that Ic(G) ≅ Ic(H) and G is cyclic, then H is cyclic. We show that for two cyclic groups G and H of orders p α1 1 . . . p αt t and q β1 1 . . . q βs s , respectively, Ic(G) ≅ Ic(H) if and only if t = s and by a suitable σ, αi = βσ(i) . Also for any cyclic groups G, H, if Ic(G) ≅ Ic(H), then →Ic(G) ≅ →Ic(H). We also show that for two finite abelian groups G and H, Ic(G) ≅ Ic(H) if and only if |π(G)| = |π(H)| and by a convenient permutation the graph of their sylow subgroups are isomorphic. In this case, their directed inclusion graphs are isomorphic too.