Sufficiency classes of 𝐿𝐶𝐴 groups

doi:10.1090/S0002-9947-1971-0291728-1

Sufficiency classes of $\textrm {LCA}$ groups
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Trans. Amer. Math. Soc. 158 (1971), 331-338 Request permission

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

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Transfinite torsion, $p$-constituents, and splitting in locally compact Abelian groups