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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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


Author: D. L. Armacost
Journal: Proc. Amer. Math. Soc. 25 (1970), 625-629
MSC: Primary 22.20
DOI: https://doi.org/10.1090/S0002-9939-1970-0260924-6
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, <IMG WIDTH="16" HEIGHT="37" ALIGN="MIDDLE" BORDER="0" SRC="images/img3.gif" ALT="$p$">-adic integers, <IMG WIDTH="16" HEIGHT="37" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="$p$">-adic numbers
Article copyright: © Copyright 1970 American Mathematical Society