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The word “group” will mean an Abelian group. As usual, A is a torsion group (a torsion-free group) if t(A) = A (t(A) = 0), where t(A) is the subgroup of all elements of finite order in A. A nonzero group A is a torsion-free group of rank 1 if it is isomorphic to a subgroup of Q; we say that this group is a rank 1 torsion-free group of idempotent type if A is isomorphic to the additive group of some subring of Q. Let A ∈ X, where X is some class of groups. We say that A is determined by its automorphism group in the class X if AutA ∼= AutB implies A ∼= B for every B ∈ X. It was proved that a p-group is determined by its automorphism group in the class of all p-groups for p > 5 [1]. Similar result was obtained for p = 3 [2]. Rank 1 torsion-free groups are not determined by their automorphism groups, but for direct sums of torsion-free groups of rank 1 (i.e., for completely decomposable torsion-free groups) necessary and sufficient conditions for determinability of a group by its automorphism group in some classes were found (see [3, 4]). Among other things, the question of determinability of a group by its automorphism group was considered in the class of completely decomposable torsion-free groups whose direct summands of rank 1 have idempotent types [3]. In particular it was shown that a torsion-free group A of rank 1 and of idempotent type is determined by its automorphism group in the class of all such groups if and only if A is an infinite cyclic group. Let us note that the class of rank 1 torsion-free groups of idempotent type coincides with the class of quotient divisible groups of rank 1 which are torsion-free. Let us denote the set of all primes by P. Recall (for example, see [5]) that a group A is p-divisible if the set pA = {pa | a ∈ A} coincides with A; a divisible group is a group which is p-divisible for all p ∈ P.

On Determinability of a Quotient Divisible Abelian Group of Rank 1 by Its Automorphism Group