Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Sufficiency classes of $ {\rm LCA}$ groups


Journal: Trans. Amer. Math. Soc. 158 (1971), 331-338
MSC: Primary 43A40
DOI: https://doi.org/10.1090/S0002-9947-1971-0291728-1
MathSciNet review: 0291728
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: By the sufficiency class $ S(H)$ of a locally compact Abelian (LCA) group $ H$ we shall mean the class of LCA groups $ G$ having sufficiently many continuous homomorphisms into $ H$ to separate the points of $ G$. 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 $ p$-groups containing dense divisible subgroups. We conclude by giving a necessary condition for two LCA groups to have the same sufficiency class.


References [Enhancements On Off] (What's this?)

  • [1] 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)
  • [2] 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)
  • [3] M. Moskowitz, Homological algebra in locally compact abelian groups, Trans. Amer. Math. Soc. 127 (1967), 361-404. MR 35 #5861. MR 0215016 (35:5861)
  • [4] L. Robertson, Connectivity, divisibility and torsion, Trans. Amer. Math. Soc. 128 (1967), 482-505. MR 36 #302. MR 0217211 (36:302)
  • [5] -, Transfinite torsion, $ p$-constituents, and splitting in locally compact Abelian groups, University of Washington (duplicated notes).

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 43A40

Retrieve articles in all journals with MSC: 43A40


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0291728-1
Keywords: Locally compact Abelian, generalized character, sufficiency class, dual sufficiency class, divisible, reduced, torsion-free, compact and connected, topological $ p$-group, torsion subgroup
Article copyright: © Copyright 1971 American Mathematical Society

American Mathematical Society