ON A COMBINATORIAL PROBLEM IN GROUP-THEORY

We say that a group G is an element of DS if for some integer m, all s ubsets X of G of size m satisfy \X(2)\ < \X\(2), where X(2) = {xy\ x, y is an element of X}. It is shown, using a previous result of Peter N eumann, that G is an element of DS if and only if either the subgroup of G generated by the squares of elements of G is finite, or G contain s a normal abelian subgroup of finite index, on which each element of G acts by conjugation either as the identity automorphism or as the in verting automorphism.