Some properties of nilpotent groups

Abstract

Property S, a finiteness property which can hold in infinite groups, was introduced by Stallings and others and shown to hold in free groups. In [2] it was shown to hold in nilpotent groups as a consequence of a technical result of Mal'cev. In that paper this technical result was dubbed property R. Hence, more generally, any property R group satisfies property S. In [7] it was shown that property R implies the following (labeled there weak property R) for a group G:

If \(G_{0}\) is any subgroup in \(G\) and \(G_{0}^{\ast}\) is any homomorphic image of \(G_{0}\), then the set of torsion elements in \(G_{0}^{\ast}\) forms a locally finite subgroup.

It was left as an open question in [7] whether weak property R is equivalent to property R. In this paper we give an explicit counterexample thereby proving that weak property R is strictly weaker than property R.