Finite subgroups of division algebras over local fields
type: Item
instance of: scholarly article
title: Finite subgroups of division algebras over local fields
zbMATH Open document ID: 0829.16023
author: Yanyan Li
published in: Journal of Algebra
publication date: 1995-11-12T00:00:00Z
review text: Let \(F\) be a complete local field of characteristic zero with finite residue field of characteristic \(p\). In the paper the finite subgroups of the multiplicative group of a division algebra \(D\) with center \(F\) are studied. The study of the finite subgroups falls naturally into two cases, the case \(p=2\) and the case \(p\) is odd. The main results describe the finite subgroups in both cases. For instance it is shown that if \(p\) is odd, all nonabelian finite subgroups of division algebras over \(F\) are isomorphic to metacyclic groups; \(G_{m,r}=\langle A,B:A^m=1\), \(BAB^{-1}=A^r\), \(B^n=A^t\rangle\) where \(n=o(r|m)\), \(t=m/(r-1,m)\). Necessary and sufficient conditions are given for \(G_{m,r}\) to embed in a division algebra over \(F\). Let \(D(F,a/b)\) be the division algebra with Hasse invariant \(a/b\in\mathbb{Q}\). Then it follows from the results of the paper that the finite subgroups of \(D(F,a/b)^*\) depend only on \(F\) and \(b\).
reviewed by: Jan Van Geel
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 11S45
zbMATH DE Number: 763959
zbMATH Keywords: multiplicative groups
zbMATH Keywords: finite subgroups of division algebras
zbMATH Keywords: metacyclic groups
zbMATH Keywords: Hasse invariant
MaRDI profile type: Publication
full work available at URL: jabr.1995.1101
OpenAlex ID: W2084852833
DOI: 10.1006/JABR.1995.1101
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description: scientific article; zbMATH DE number 763959