Normalizers of finite subgroups of division algebras over local fields
type: Item
instance of: scholarly article
title: Normalizers of finite subgroups of division algebras over local fields
zbMATH Open document ID: 0958.16034
author: Thomas Hewett
published in: Mathematical Research Letters
publication date: 2000-08-13T00:00:00Z
review text: Let \(D\) be a central division algebra over a local field \(F\) of characteristic \(0\) with finite residue class field of odd characteristic \(p\). The author [J. Algebra 173, No. 3, 518-548 (1995; Zbl 0829.16023)] proved that any finite non-Abelian subgroup of \(D\) has the form \(G_{m,r}=gp(A,B\mid A^m=1,\;BAB^{-1}=A^r,\;B^e=A^t)\), where \(r\in\mathbb{Z}/m\mathbb{Z}\) has order \(e\) and \(t=m/(r-1,m)\), and he gave conditions for such \(G_{m,r}\) to occur. In the present paper he makes a study of the normalizer of these groups. He first describes the automorphism group associated with different embeddings of \(G_{m,r}\) and relates it to the valuations on \(F\). He also briefly examines the case of cyclic groups and the case of residue characteristic 2.
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 11S45
Mathematics Subject Classification ID: 12E15
zbMATH DE Number: 1489514
zbMATH Keywords: central division algebras
zbMATH Keywords: local fields
zbMATH Keywords: finite non-Abelian subgroups
zbMATH Keywords: normalizers
zbMATH Keywords: automorphism groups
zbMATH Keywords: cyclic groups
reviewed by: Paul M. Cohn
MaRDI profile type: Publication
full work available at URL: mrl.1999.v6.n3.a2
OpenAlex ID: W2314178967
DOI: 10.4310/MRL.1999.V6.N3.A2
Recommended article: Finite subgroups of division algebras over local fields
Recommended article: Q4613670
Recommended article: The normality of locally finite associative division algebras over classical fields
Recommended article: Normalizers of subgroups of division rings
Recommended article: On division subrings normalized by almost subnormal subgroups in division rings
Recommended article: NORMALIZERS OF NILPOTENT SUBGROUPS OF DIVISION RINGS
Recommended article: Normal Division Algebras Over a Modular Field
Recommended article: Pure, normal maximal subfields for division algebras in the Schur subgroup
description: scientific article; zbMATH DE number 1489514