Some properties of nilpotent groups

A. M. Gaglione ; S. Lipschutz ; D. Spellman

Algebra and discrete mathematics, no. 4 (2009), pp. 66-77

Résumé

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 propert 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 $G_{0}^{*}$ is any homomorphic image of $G_{0}$, then the set of torsion elements in $G_{0}^{*}$ 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.

Détail

Keywords: Property S, Property R, variety of groups, closure operator.
Mots-clés : commensurable