proceedings of the american mathematical society Volume 117, Number 3, March 1993 A CHARACTERIZATION OF ABELIAN GROUPS L. BRAILOVSKY (Communicated by Ronald M. Solomon) Abstract. Let G be a group and let k > 2 be an integer such that (fc3 - k) < \G\/2 if G is finite. Suppose that the condition \A2\< k(k + l)/2 is satisfied by every fc-element subset A C G. Then G is abelian. 1. Introduction The problem considered in this paper comes from a class of problems con- cerning the structure of groups that satisfy the small squaring property on 7c-sets. This notion was introduced in [BFP] and is as follows. Let G be a group and 7c > 1 an integer. For a finite subset A c G set A2 = {aiOj | ai, aj £ A}. We say that G has the small squaring property on A>sets (and we write G £ DS(k)) if \A2\ < k2 for any subset A c G of order k. Obviously, if G is abelian then G £ DS(k) for any k > 1, and the small squaring property may be regarded as a generalization of the commutativity property. The classification of nonabelian DS(k) groups was performed for k - 2 in [Fr]. For k = 3 the classification was basically done in [BFP], and it was completed and generalized for infinite groups in [LM]. For the general k it was shown by Neumann (see [HLM]) that if G £ DS(k) then G has normal subgroups M, N with 1 < M < N < G such that \M\ and \G/N\ are bounded above by a function of k and N/M is abelian, i.e., G is "finite by abelian by finite". This result was used in [HLM] to show that a group G £ DS(k) if and only if G is either "abelian by finite" or "finite by elementary abelian 2-group". However, if G is abelian then not only G £ DS(k) but also \A2\<k(k+l)/2 for any subset A C G of order k . In this paper we show that this "very small" squaring property actually characterizes abelian groups. Received by the editors July 1, 1991. 1991 Mathematics Subject Classification. Primary 20A05, 20E34. The content of this paper corresponds to a part of the author's Ph.D. thesis research carried out in Tel-Aviv University under the supervision of Professors G. Freiman and M. Herzog. © 1993 American Mathematical Society 0002-9939/93 $1.00+ $.25 per page 627 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use