Examples of locally compact noncompact minimal topological groups

Susanne Dierolf and Ulrich Schwanengel

Vol. 82 (1979), No. 2, 349–355

DOI: 10.2140/pjm.1979.82.349

Abstract

In 1971, R. M. Stephenson, Jr., [4], showed that an abelian locally compact topological group must be compact if it is minimal (i.e., if it does not admit a strictly coarser Hausdorff group topology). He left open the question, whether there exist locally compact noncompact minimal topological groups.

In this note we give an example of a closed noncompact subgroup of GL (2;R) which is minimal. Moreover we prove that every discrete topological group is topologically isomorphic to a subgroup of a locally compact minimal topological group. Another example shows that a minimal topological group can contain a discrete, nonminimal normal subgroup.

Mathematical Subject Classification 2000

Primary: 22D05

Milestones

Received: 23 August 1978

Published: 1 June 1979