R̄-Groups

We introduce a new class of torsion-free groups, calledR̄-groups, in which the normalizer and the centralizer of the isolator of each cyclic subgroup coincide. This is an extensive class of groups containing all torsion-free locally nilpotent groups and all free groups. These groups share many interesting properties with the class of allR-groups, groups having unique roots. For example, we show that the class ofR̄-groups is closed under products, subgroups, restricted wreath products, free products, and certain free products with amalgamation. Because of these nice properties, the class induces well behaved closure operators and we investigate the concomitant categorically compact classes associated with these operators. These new compact classes are closed under finite products and homomorphic images and their characterization generally involves the existence of roots of all orders for elements of the hypercenter.