On a combinatorial problem in group theory

Abstract

We say that a groupG ∈DS if for some integerm, all subsetsX ofG of sizem satisfy |X 2|<|X|2, whereX 2={xy|x,y ∈X}. It is shown, using a previous result of Peter Neumann, thatG ∈DS if and only if either the subgroup ofG generated by the squares of elements ofG is finite, orG contains a normal abelian subgroup of finite index, on which each element ofG acts by conjugation either as the identity automorphism or as the inverting automorphism.