The finite quotients of the multiplicative group of a division algebra of degree \(3\) are solvable
type: Item
instance of: scholarly article
title: The finite quotients of the multiplicative group of a division algebra of degree \(3\) are solvable
zbMATH Open document ID: 0938.16024
DOI: 10.1007/BF02810692
published in: Israel Journal of Mathematics
publication date: 1999-12-06T00:00:00Z
review text: Let \(D\) be a division algebra of finite degree \(n\) over its centre \(F\). Given a normal subgroup \(N\) of \(D^*=D\setminus\{0\}\) containing \(F^*\) and \(H=D^*/N\), the authors show that the centre of \(H\) (and hence \(H/H'\)) has exponent dividing \(n\). They further prove that when \(n=3\), the finite quotients of \(D^*\) are soluble. The proof uses the factorization theorem of \textit{J. H. M. Wedderburn} [Trans. Am. Math. Soc. 22, 129-135 (1921; JFM 48.0126.01)], which states that the right zeros of \(fg\) are right zeros of \(g\) or conjugates of right zeros of \(f\).
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 16K20
zbMATH DE Number: 1356484
zbMATH Keywords: division algebras of finite degree
zbMATH Keywords: normal subgroups
zbMATH Keywords: centre
zbMATH Keywords: exponent
zbMATH Keywords: finite quotients
zbMATH Keywords: factorization theorem
zbMATH Keywords: JFM 48.0126.01
zbMATH Keywords: right zeros
author: Louis Rowen
author: Yoav Segev
reviewed by: Paul M. Cohn
MaRDI profile type: Publication
cites work: Rational identities and applications to algebra and geometry
cites work: Q3708174
cites work: Q4835117
cites work: Q4692288
cites work: Q4011058
cites work: Q4854353
cites work: Q5331572
cites work: On finite homomorphic images of the multiplicative group of a division algebra
cites work: Anisotropic groups of type \(A_n\) and the commuting graph of finite simple groups.
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description: scientific article; zbMATH DE number 1356484