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 | References | Similar Articles | Additional Information
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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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), 225β235. 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), 1453β1456. 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), 241β244. MR 171878, DOI https://doi.org/10.4064/fm-56-2-241-244
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Additional Information
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