Abstract
For a class of groupsF, closed under formation of subgroups and products, we call a subgroupA of a groupG F-regular provided there are two homomorphismsf, g: G » F, withF εF, so thatA = {x εG |f(x) =g(x)}.A is calledF-normal providedA is normal inG andG/A εF. For an arbitrary subgroupA ofG, theF-regular (respectively,F-normal) closure ofA inG is the intersection of allF-regular (respectively,F-normal) subgroups ofG containingA. This process gives rise to two well behaved idempotent closure operators.
A groupG is calledF-regular (respectively,F-normal) compact provided for every groupH, andF-regular (respectively,F-normal) subgroupA ofG × H, π2(A) is anF-regular (respectively,F-normal) subgroup ofH. This generalizes the well known Kuratowski-Mrówka theorem for topological compactness.
In this paper, theF-regular compact andF-normal compact groups are characterized for the classesF consisting of: all torsion-free groups, allR-groups, and all torsion-free abelian groups. In doing so, new classes of groups having nice properties are introduced about which little is known.
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Fay, T.H., Walls, G.L. Regular and normal closure operators and categorical compactness for groups. Appl Categor Struct 3, 261–278 (1995). https://doi.org/10.1007/BF00878444
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DOI: https://doi.org/10.1007/BF00878444