On Deficient Squares Groups and Fully‐Independent Subsets

In 1976, B. H. Neumann [4] characterized the central-by-finite groups by the following property: the group G is central-by-finite if and only if it does not contain an infinite independent subset, where a subset U of G is called independent if uv = vu for u,veUimplies u = v. More recently, P. Longobardi, M. Maj and the first author [2] considered a more general class of groups, the deficient squares groups. A group is said to satisfy the deficient squares property if there exists an integer m such that \M*\ < |A/j for all subsets M of G of size m, where M = {mn \m,ne M). They showed that a group G satisfies the deficient squares property if and only if one of the following holds: G is central-by-finite, |{g|ge(j}| is finite, or G is nearly dihedral, where by nearly dihedral we mean a group G with an abelian normal subgroup H of finite index, on which each element of G acts by conjugation either as the identity automorphism or as the inverting automorphism. The aim of this paper is to characterize the deficient squares groups by a property of its infinite subsets, in the spirit of the above-mentioned result of B. H. Neumann. A subset U of a group G will be called fully-independent if uv = xy for u,v, x,ye U implies u = x and v = y. We prove that a group G satisfies the deficient squares property if and only if it does not contain an infinite fully-independent subset. We state and prove our results in the opposite direction. Thus our main result is the following.