EVERY ABELIAN GROUP IS DETERMINED BY A SUBGROUP LATTICE

We prove that every Abelian group G is determined up to an isomorphism by the subgroup lattice of the group Z×G and some other similar results. It is well known that the lattice L(G) of all subgroups of a group G does not determine the group, that is, there exist non-isomorphic groups with isomorphic subgroup lattices. However, sometimes, specific subgroup lattices may determine some groups. For example, it was proved in [2] that every Abelian group which has the square root property is determined by the subgroup lattice of its square. This result generalizes an earlier result of Lukács and Pálfy [4]. The main result of this note is the following theorem which actually can be proved in a more general setting (see Lemma 2). Theorem 1. Let A be an Abelian group and G a group. The following statements are true. (i) If Z×A and Z×G have isomorphic subgroup lattices then A ∼ = G. (ii) If Q×A and Q×G have isomorphic subgroup lattices then A ∼ = G. (iii) If A is an Abelian p-group and G is a p-group such that Z(p∞)×A and Z(p∞)×G have isomorphic subgroup lattices then A ∼ = G