Minimality of the Semidirect Product

Abstract

A topological group is minimal if it does not admit a strictly coarser Hausdorff group topology. We provide a sufficient and necessary condition for the minimality of the semidirect product G ⋋ P, where G is a compact topological group and P is a topological subgroup of Aut(G). We prove that G ⋋ P is minimal for every closed subgroup P of Aut(G). In case G is abelian, the same is true for every subgroup P ⊆ Aut(G). We show, in contrast, that there exist a compact two-step nilpotent group G and a subgroup P of Aut(G) such that G ⋋ P is not minimal. This answers a question of Dikranjan. Some of our results were inspired by a work of Gamarnik [12].