"Regular and Normal Closure Operators and Categorical Compactness for G" by Temple H. Fay and Gary L. Walls
 

Regular and Normal Closure Operators and Categorical Compactness for Groups

Document Type

Article

Publication Date

9-1-1995

Department

Mathematics

School

Mathematics and Natural Sciences

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.

Publication Title

Applied Categorical Structures

Volume

3

Issue

3

First Page

261

Last Page

278

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