Minimality of the semidirect product

Michael Megrelishvili, Luie Polev, Menachem Shlossberg

Research output: Contribution to journalArticlepeer-review


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 GλP; where G is a compact topological group and P is a topological subgroup of Aut(G). We prove that GλP 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 GλP is not minimal. This answers a question of Dikranjan. Some of our results were inspired by a work of Gamarnik [12].

Original languageEnglish
Pages (from-to)199-210
Number of pages12
JournalTopology Proceedings
StatePublished - 2017

Bibliographical note

Publisher Copyright:
© 2016 Topology Proceedings.


  • Automorphism group
  • Minimal topological group
  • Ordered space
  • Semidirect product


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