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Proceedings of the American Mathematical Society

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Well-known $ {\rm LCA}$ groups characterized by their closed subgroups

Author: D. L. Armacost
Journal: Proc. Amer. Math. Soc. 25 (1970), 625-629
MSC: Primary 22.20
MathSciNet review: 0260924
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Abstract: In this paper we determine (1) the class of all nondiscrete $ \operatorname{LCA} $ groups for which every proper closed subgroup is the kernel of a continuous character of the group, (2) the class of locally compact groups whose closed subgroups are totally ordered by inclusion, and (3) the class of infinite $ \operatorname{LCA} $ groups whose proper closed subgroups are topologically isomorphic. Since all these determinations involve only the most common $ \operatorname{LCA} $ groups, we may regard our findings as characterizations of natural classes of these well-known groups.

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Keywords: Locally compact Abelian, closed subgroups, kernels of characters, subgroups ordered by inclusion, topologically isomorphic subgroups, real numbers, circle, integers, quasicyclic groups, $ p$-adic integers, $ p$-adic numbers
Article copyright: © Copyright 1970 American Mathematical Society

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