2015/11/22 by Megrelishvili, Michael, Polev, Luie, Shlossberg, Menachem
#FOS: Mathematics #General Topology (math.GN) #Group Theory (math.GR)
paper · doi:10.48550/arxiv.1511.07021
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\leftthreetimes P, where G is a compact topological group and P is a topological subgroup of Aut(G). We prove that G\leftthreetimes 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\leftthreetimes P is not minimal. This answers a question of Dikranjan. Some of our results were inspired by a work of Gamarnik.