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Commutators in pseudo-orthogonal groups

Part of: Linear algebraic groups and related topics Other groups of matrices

Published online by Cambridge University Press:  09 April 2009

F. A. Arlinghaus ,
L. N. Vaserstein  and
Hong You
F. A. Arlinghaus
Affiliation:
Department of Mathematics YoungstownState University Youngstown, Ohio 44455, USA
L. N. Vaserstein
Affiliation:
Department of Mathematics The PennsylvaniaState University University Park, Pennsylvania 16802USA e-mail: [email protected]
Hong You
Affiliation:
Department of Mathematics Northeast NormalUniversity Changchun130024 People's Republic of China
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Abstract

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We study commutators in pseudo-orthogonal groups O2nR (including unitary, symplectic, and ordinary orthogonal groups) and in the conformal pseudo-orthogonal groups GO2nR. We estimate the number of commutators, c(O2nR) and c(GO2nR), needed to represent every element in the commutator subgroup. We show that c(O2nR) ≤ 4 if R satisfies the ∧-stable condition and either n ≥ 3 or n = 2 and 1 is the sum of two units in R, and that c(GO2nR) ≤ 3 when the involution is trivial and ∧ = R∈. We also show that c(O2nR) ≤ 3 and c(GO2nR) ≤ 2 for the ordinary orthogonal group O2nR over a commutative ring R of absolute stable rank 1 where either n ≥ 3 or n = 2 and 1 is the sum of two units in R.

MSC classification

Secondary: 20G35: Linear algebraic groups over adèles and other rings and schemes 20H25: Other matrix groups over rings
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 59 , Issue 3 , December 1995 , pp. 353 - 365
Copyright
Copyright © Australian Mathematical Society 1995

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