REGULAR AND NORMAL CLOSURE OPERATORS AND CATEGORICAL COMPACTNESS FOR GROUPS

For a class of groups F, closed under formation of subgroups and products, we call a subgroup A of a group G F-regular provided there are two homomorphisms f,g: G --> F, with F is an element of F, so that A = {x is an element of G \ f(x) = g(x)}. A is called F-normal provided A is normal in G and G/A is an element of F. For an arbitrary subgroup A of G, the F-regular (respectively, F-normal) closure of A in G is the intersection of all F-regular (respectively, F-normal) subgroups of G containing A. This process gives rise to two well behaved idempotent closure operators. A group G is called F-regular (respectively, F-normal) compact provided for every group H, and F-regular (respectively, T-normal) subgroup A of G x H, pi(2)(A) is an F-regular (respectively, F-normal) subgroup of H. This generalizes the well known Kuratowski-Mrowka theorem for topoiogical compactness. In this paper, the F-regular compact and F-normal compact groups are characterized for the classes F consisting of: all torsion-free groups, all R-groups, and all torsion-free abelian groups. In doing so, new classes of groups having nice properties are introduced about which little is known.