Minimality of the Semidirect Product

Michael Megrelishvili, Luie Polev, Menachem Shlossberg

Published 2015-11-22Version 1

A topological group is minimal if it does not admit a strictly coarser Hausdorff group topology. We prove that for a compact topological group $G$, the semidirect product $G\leftthreetimes P$ is minimal for every closed subgroup $P$ of $Aut(G)$. In general, the compactness of $G$ is essential; $G\leftthreetimes P$ might be nonminimal even for precompact minimal groups $G$ as it follows from an example of Eberhardt-Dierolf-Schwanengel. Some of the results were inspired by a work of Gamarnik .