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 Chachólski, Parent and Stanley in 2004. We also establish several new relations between singly-generated closed classes. IntroductionIn a category of R-modules over a certain ring R, the study of the closure of different classes under different operations has been a hot research topic for decades (see [16] for a thorough approach). The operations include extensions, subgroups, quotients, etc. In the case of R = Z, we are dealing, of course, with operations in the category of abelian groups.There is another different but related problem. Given a collection D of objects in a class C, the smallest class that contains D and is closed under certain operations is called the class generated by D under these operations. When this class is equal to C, it is said that C is generated by D under the operations, and the elements of D are called generators of C. In this situation, if D consists of only one object, the class C is said to be