The \(A\)-core and \(A\)-cover of a group. (Q1012580)

This paper is an introduction to, and thorough investigation of, the concept of cellular approximation in the context of group theory. The notion originally arose in topology but has subsequently been applied in several other categories. If \(A\) is a group, then \(A\)-cellularity is the study of groups as seen by \(A\). More precisely, it is concerned with group homomorphisms out of \(A\). For example, a group homomorphism \(f\colon G\to H\) is an \(A\)-equivalence if composition with \(f\) gives a bijection \(\Hom(A,G)\to\Hom(A,H)\). A group \(C\) is said to be \(A\)-cellular if any \(A\)-equivalence is a \(C\)-equivalence. The \(A\)-cover of a group \(G\) is an \(A\)-cellular group which is \(A\)-equivalent to \(G\). This is to be thought of as the best ``approximation'' to \(G\) that can be ``built'' from \(A\). Many related notions are defined and studied, and some connections with the Schur multiplier are described.