Compact groups having almost discrete orbit hypergroups
type: Item
instance of: scholarly article
title: Compact groups having almost discrete orbit hypergroups
zbMATH Open document ID: 0882.43008
DOI: 10.1007/BF01320187
author: Michael Voit
published in: Monatshefte für Mathematik
publication date: 1997-09-28T00:00:00Z
full work available at URL: https://eudml.org/doc/178745
review text: Let \(G\) be a compact group of automorphisms acting continuously on a compact group \(H\), let \(H^G\) be the space of all \(G\)-orbits in \(H\) with natural hypergroup structure, let \(\mathbb{Z}(p)\) be a cyclic group and \(\mathbb{Z}_p\) the group of \(p\)-adic integers (\(p\) a prime). The author proves: (1) Let \(H\) be a compact group such that the orbit hypergroup \(H^G\) is almost discrete for some compact subgroup \(G\) of \(\Aut(H)\). If \(H\) contains an open solvable subgroup \(L\), then \(H\) is isomorphic either to \(\mathbb{Z}^n_p\) or to \(\mathbb{Z}(p)^N\) for some prime \(p\in N\). (2) Let \(H=\mathbb{Z}^n_p\), then \(H^G\) is almost discrete if and only if \(G\) is essentially an open subgroup of \(\text{CL}_n(\mathbb{Z}_p)\). -- Some interesting examples are also considered.
Mathematics Subject Classification ID: 43A62
Mathematics Subject Classification ID: 20N20
zbMATH DE Number: 1051735
zbMATH Keywords: compact group
zbMATH Keywords: \(G\)-orbits
zbMATH Keywords: hypergroup
zbMATH Keywords: orbit hypergroup
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1051735