Sufficiency classes of groups
Abstract: By the sufficiency class of a locally compact Abelian (LCA) group we shall mean the class of LCA groups having sufficiently many continuous homomorphisms into to separate the points of . In this paper we determine the sufficiency classes of a number of LCA groups and indicate how these determinations may help to describe the structure of certain classes of LCA groups. In particular, we give a new proof of a theorem of Robertson which states that an LCA group is torsion-free if and only if its dual contains a dense divisible subgroup. We shall also derive some facts about the compact connected Abelian groups and a result about topological -groups containing dense divisible subgroups. We conclude by giving a necessary condition for two LCA groups to have the same sufficiency class.
-  Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR 551496
-  Gerald L. Itzkowitz, The existence of homomorphisms in compact connected Abelian groups, Proc. Amer. Math. Soc. 19 (1968), 214–216. MR 0219655, https://doi.org/10.1090/S0002-9939-1968-0219655-1
-  Martin Moskowitz, Homological algebra in locally compact abelian groups, Trans. Amer. Math. Soc. 127 (1967), 361–404. MR 0215016, https://doi.org/10.1090/S0002-9947-1967-0215016-3
-  Lewis C. Robertson, Connectivity, divisibility, and torsion, Trans. Amer. Math. Soc. 128 (1967), 482–505. MR 0217211, https://doi.org/10.1090/S0002-9947-1967-0217211-6
-  -, Transfinite torsion, -constituents, and splitting in locally compact Abelian groups, University of Washington (duplicated notes).
- E. Hewitt and K. Ross, Abstract harmonic analysis. Vol. 1, Structure of topological groups. Integration theory, group representation, Die Grundlehren der math. Wissenschaften, Band 115, Academic Press, New York; Springer-Verlag, Berlin, 1963. MR 28 #158. MR 551496 (81k:43001)
- G. L. Itzkowitz, The existence of homomorphisms in compact connected Abelian groups, Proc. Amer. Math. Soc. 19 (1968), 214-216. MR 36 #2734. MR 0219655 (36:2734)
- M. Moskowitz, Homological algebra in locally compact abelian groups, Trans. Amer. Math. Soc. 127 (1967), 361-404. MR 35 #5861. MR 0215016 (35:5861)
- L. Robertson, Connectivity, divisibility and torsion, Trans. Amer. Math. Soc. 128 (1967), 482-505. MR 36 #302. MR 0217211 (36:302)
- -, Transfinite torsion, -constituents, and splitting in locally compact Abelian groups, University of Washington (duplicated notes).
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Keywords: Locally compact Abelian, generalized character, sufficiency class, dual sufficiency class, divisible, reduced, torsion-free, compact and connected, topological -group, torsion subgroup
Article copyright: © Copyright 1971 American Mathematical Society