On pure subgroups of locally compact abelian groups (Q1423785)

A subgroup \(H\) of a group \(G\) is called \(pure\) (\(in\) \(G\)) if for every positive integer \(n\) and every \(h\in H\), the equation \(nx=h\) is solvable in \(H\) whenever it is solvable in \(G\). An LCA group \(G\) \(has\) \(correct\) \(purity\) if a closed subgroup of \(G\) is pure exactly if its annihilator is pure (in the Pontrjagin dual \(\hat G\)). A topological group is said to have \(no\) \(small\) \(subgroups\) if there is a neighborhoood of zero which does not contain any nontrivial subgroups. \textit{S. Hartman} and \textit{A. Hulanicki} [Fundam. Math. 45, 71--77 (1957; Zbl 0083.25501)] have shown that LCA groups that are either compactly generated or have no small subgroups have correct purity. They asked if the same can be said about LCA groups of the form \(C\times D\), where \(C\) is either compactly generated or has no small subgroups, and \(D\) is discrete. In the present paper this question is answered negatively. The author gives an example of an LCA group \(G=C\times D\) (where \(C\) is a compact and \(D\) is a discrete group) and a closed pure subgroup of \(G\) having nonpure annihilator in \(\hat G\).