Generators and closed classes of groups

Abstract

We show that, in the category of groups, every singly-generated class which is closed under isomorphisms, direct limits, and extensions is also singly-generated under isomorphisms and direct limits, and in particular is co-reflective. In this way, we extend to the group-theoretic framework the topological analogue proved by Chach´olski, Parent, and Stanley in 2004. We also establish several new relations between singly-generated closed classes.