Well-known 𝐿𝐶𝐴 groups characterized by their closed subgroups

Armacost, D. L.

doi:10.1090/S0002-9939-1970-0260924-6

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.

E. Hewitt and K. Ross, Abstract harmonic analysis, Vol. 1, Die Grundlehren der math. Wissenschaften, Band 115, Academic Press, New York and Springer-Verlag, Berlin and New York, 1963. MR 28 #158.

Irving Kaplansky, Infinite abelian groups, University of Michigan Press, Ann Arbor, 1954. MR 0065561
Neil W. Rickert, Locally compact topologies for groups, Trans. Amer. Math. Soc. 126 (1967), 225–235. MR 202911, DOI https://doi.org/10.1090/S0002-9947-1967-0202911-4
Lewis Robertson and Bert M. Schreiber, The additive structure of integer groups and $p$-adic number fields, Proc. Amer. Math. Soc. 19 (1968), 1453–1456. MR 230836, DOI https://doi.org/10.1090/S0002-9939-1968-0230836-3
K. A. Ross, Closed subgroups of locally compact Abelian groups, Fund. Math. 56 (1964), 241–244. MR 171878, DOI https://doi.org/10.4064/fm-56-2-241-244