Sufficiency classes of $\textrm {LCA}$ groups

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.