Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-25T09:23:06.506Z Has data issue: false hasContentIssue false
  • English
  • Français

Commutators in pseudo-orthogonal groups

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

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[1]Bak, A., K-theory of forms (Princeton University Press, Princeton, 1981).Google Scholar
[2]Bass, H., ‘K-theory and stable algebra’, Publ. Math. IHES 22 (1964), 5–60.CrossRefGoogle Scholar
[3]Bass, H., ‘Unitary algebraic K-theory’, in: Lecture Notes in Math., 343 (Springer, Berlin, 1973) pp. 57–265.Google Scholar
[4]Dennis, R. K. and Vaserstein, L. N., ‘On a question of M. Newman on the number of commutators’, J. Algebra 118 (1988), 150–161.CrossRefGoogle Scholar
[5]Magurn, B. A., van der Kallen, W. and Vaserstein, L. N., ‘Absolute stable rank and Witt cancellation for noncommutative rings’, Inv. Math. 91 (1988), 525–542.CrossRefGoogle Scholar
[6]Vaserstein, L. N., ‘On normal subgroups of Chevalley groups over commutative rings’, Tohôku Math. J. 38 (1986), 219–230.Google Scholar
[7]Vaserstein, L. N. and Wheland, E., ‘Commutators and companion matrices over rings of stable rank 1’, Linear Algebra and its Applications 142 (1990), 263–277.CrossRefGoogle Scholar
[8]You, H., ‘Commutators and unipotents in symplectic groups’, Acta Math. Sinica 10 (1994), 173–179.Google Scholar