The A-core and A-cover of a group

This paper provides a comprehensive investigation of the cellular approximation functor cellAG, in the category of groups, approximating a group G by a group A. We also study related notions such as A-injection, A-generation and A-constructibility of a group G and we find several interesting connections with the Schur multiplier H2(G,Z). Our constructions are direct and are given in a slow and detailed manner.