On the extraction of roots in exponential $A$-groups

Abstract

An exponential A-group is a group which comes equipped with an A-action (A is a commutative ring with unity), satisfying certain axioms. In this paper, we investigate some aspects of root extraction in the category of exponential A-groups. Of particular interest is the extraction of roots in nilpotent R-powered groups. Among other results, we prove that if R is a PID and G is a nilpotent R-powered group for which root extraction is always possible, then the torsion R-subgroup of G lies in the center. Furthermore, if the torsion R-subgroup is finitely R-generated, then G is torsion-free.