Compact nilpotent groups |
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Authors: | Temple H Fay Gary L Walls |
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Institution: | University of Southern Mississippi , Hattiesburg, Mississippi, 39406 |
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Abstract: | Herrlich, Salicrup, and Strecker HSS] have shown that Kuratowski’s Theorem, namely, that a space X is compact if and only if for every space Y, the projection π2X×Y → Y is a closed map, can be interpreted categorically, and hence generalized and applied in a wider settin than the category of topological spaces. The first author, in an earlier paperj Fl] , applied this categorical interpretation of compactness in categories of R-modules, obtaining a theory of compactness for each torsion theory T. In the case of the category of abelian groups and a hereditary torsion theory T, a group G is T-compact provided G/TG is a T-injective. In this note, the notion of compact is extended to the categories of hypercentral groups, nilpotent groups, and of FC-groups; it is shown that if T π denotes the π-torsion subgroup functor for a set of primes π, then a group G is T π-compact provided G/T πG is π-complete, extending the abelian group result in a natural way. |
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