Regular and normal closure operators and categorical compactness for groups (Q1899875)

The authors introduce, for each epireflective subcategory \(\mathbf A\) of the category \textbf{Grp} of groups, two closure-operators: the \(\mathbf A\)- regular closure and the \(\mathbf A\)-normal closure. Further, each closure operator \(c\) on \(\mathbf{Grp}\) allows one to define the concept of \(c\)-compactness: a group \(G\) is \(c\)-compact provided that for any group \(H\) the projection \(G \times H \to H\) preserves \(c\)-closed subgroups. Combining the above two steps, the authors obtain for each \(\mathbf A\) as above the concepts of \(\mathbf A\)-regular compact and of \(\mathbf A\)-normal compact groups. The authors investigate these concepts for general \(\mathbf A\) and for the following special cases: \({\mathbf A}_1\) consists of all torsion-free groups, \({\mathbf A}_2\) of all \(R\)-groups, and \({\mathbf A}_3\) of all torsion-free abelian groups. Typical results are: 1. A group is \(\mathbf A\)-regular (resp. \(\mathbf A\)-normal) compact if and only if its \(\mathbf A\)-reflection is \(\mathbf A\)-regular (resp. \(\mathbf A\)-normal) compact. 2. For varieties \(\mathbf A\) every group is \(\mathbf A\)-normal compact. 3. Every group is \({\mathbf A}_1\) regular compact. 4. Equivalent are: (a) \(G\) is \({\mathbf A}_3\)-regular compact, (b) \(G\) is \({\mathbf A}_3\)-normal compact, (c) the \({\mathbf A}_3\)-reflection of \(G\) is divisible, (d) every torsion-free Abelian image of \(G\) is divisible.