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 and , there is an arc from to if and only if ⊂ . 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 and are adjacent if and only if ⊂ or ⊂ . 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.