Abelian Groups Determined by Subgroup Lattices of Direct Powers

In this short note, we show that the class of abelian groups determined by the subgroup lattice of their direct n-powers is exactly the class of the abelian groups which share the n-root property. As applications we answer in the negative a (semi)conjecture of Palfy and solve a more general problem. Recently, for an arbitrary group G, the subgroup lattice of the square G×G has received some attention (see [10] and [13]) with respect to the following Problem: If the direct squares of two groups are projective (i.e., have isomorphic subgroup lattices), are these groups isomorphic? In other words, does the subgroup lattice of the direct square determine the group up to an isomorphism? This problem has negative answer: the A. Rottlaender groups (see [11]). However, for simple groups (see [14]) and for finite abelian groups (see [9]) the answer is affirmative. Let S2 be the class of the abelian groups which share the square root property (i.e., G ∈ S2 iff G ∼= H whenever G ⊕ G ∼= H ⊕ H) and P2 the class of the abelian groups which are determined by the subgroup lattice of their direct squares (i.e., G ∈ P2 iff G ∼= H whenever L(G ⊕ G) ∼= L(H ⊕ H), lattice isomorphism; here L(G) denotes the subgroup lattice of the group G). For notation and terminology we refer to [5]. Main result: S2 = P2. Theorem 1. An abelian group is determined by the subgroup lattice of its direct square if and only if it has the square-root property. Proof. Actually, more can be proved: let G be an abelian group and H an arbitrary group. G×G is projective to H×H if and only if G×G and H×H are isomorphic. If G is abelian, so is G × G. Hence L(G × G) is modular and so is L(H × H). Using a result of Lukacs-Palfy, (see [9]), H must be abelian, too. If G is not torsion then the torsion-free rank r0(G×G) is at least 2. Since G×G is projective to H × H , by an early result of Baer (see [3]), G × G and H × H are isomorphic. If G is torsion then the p-rank rp(G × G) is at least 2 for every p such that the primary component Gp 6= 0. Again, since G×G is projective to H×H , by a similar result due to Fuchs (see [5]), G × G ∼= H × H . 2000 Mathematics Subject Classification. 20 K 27; 20 D 30; 06 B 25.