Cellular covers of groups

In this paper we discuss the concept of cellular cover for groups, especially nilpotent and finite groups. A cellular cover is a group homomorphism c:G→Mc:G→M such that composition with cc induces an isomorphism of sets between Hom(G,G) and Hom(G,M). An interesting example is when GG is the universal central extension of the perfect group MM. This concept originates in algebraic topology and homological algebra, where it is related to the study of localizations of spaces and other objects. As explained below, it is closely related to the concept of cellular approximation of any group by a given fixed group. We are particularly interested in properties of MM that are inherited by GG, and in some cases by properties of the kernel of the map cc.