# On Pure Subgroups of Locally Compact Abelian Groups

## Document Type

Peer-Reviewed Article

## Publication Date

9-2003

## Abstract

In this note, we construct an example of a locally compact abelian group *G* = *C* × *D* (where *C* is a compact group and *D* is a discrete group) and a closed pure subgroup of *G* having nonpure annihilator in the Pontrjagin dual $\hat{G}$, answering a question raised by Hartman and Hulanicki. A simple proof of the following result is given: Suppose ${\frak K}$ is a class of locally compact abelian groups such that $G \in {\frak K}$ implies that $\hat{G} \in {\frak K}$ and *nG* is closed in *G* for each positive integer *n*. If *H* is a closed subgroup of a group $G \in {\frak K}$, then *H* is topologically pure in *G* exactly if the annihilator of *H* is topologically pure in $\hat{G}$. This result extends a theorem of Hartman and Hulanicki.

## DOI

10.1007/s00013-003-0823-z

## Recommended Citation

Loth, P. (2003). On pure subgroups of locally compact abelian groups. *Archiv der Mathematik 81*(3), 255-257. doi: 10.1007/s00013-003-0823-z