Hostname: page-component-84b7d79bbc-x5cpj Total loading time: 0 Render date: 2024-07-26T14:23:10.244Z Has data issue: false hasContentIssue false
  • English
  • Français

Stable sandwich classification theorem for classical-like groups

Published online by Cambridge University Press:  01 November 2007

ZUHONG ZHANG
ZUHONG ZHANG*
Affiliation:
Department of Pure Mathematics, Queen's University, Belfast BT7 1NN, N. Ireland. email: zuhong.zhang@qub.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Let H denote a subgroup of the unitary group U(R, Λ) which is normalized by EU(J, ΓJ) for some form ideal (J, ΓJ) of a commutative form ring (R, Λ). We prove that H satisfies a “sandwich” property, i.e., there exists a form ideal (I ΓI) such that for some form ideal (I : J7, Ω). This answers a conjecture of Bak (1967) in the stable case.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 143 , Issue 3 , November 2007 , pp. 607 - 619
Copyright
Copyright © Cambridge Philosophical Society 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Bak, A.. The stable structure of quadratic modules. Thesis (Columbia University, 1969).Google Scholar
[2]Bak, A.. K-theory of forms. Ann. of Math. Stud. 98 (Princeton University Press, 1981).Google Scholar
[3]Bak, A.. Subgroups of the general linear group normalized by relative elementary groups, in: Algebraic K-theory, Part II. Lecture Notes in Math. 967 (1982), pp. 1–22.Google Scholar
[4]Bak, A. and Vavilov, N. A.. Structure of hyperbolic unitary groups I: elementary subgroups. Algebra Colloq. 7:2 (2000), 159–196.Google Scholar
[5]Bak, A., Petrov, V. and Tang, G.. Stability for quadratic K 1. K-Theory 30 (2003), 1–11.Google Scholar
[6]Habdank, G.. Mixed commutator groups in classical groups and a classification of subgroups of classical groups normalized by relative elementary groups. Dissertation (Universitaet Bielefeld, 1987).Google Scholar
[7]Hahn, A. J. and O'Meara, O. T.. The Classical Groups and K-Theory (Springer, 1989).Google Scholar
[8]Hazrat, R.. Dimension theory and nonstable K 1 of quadratic modules. K-theory 27 (2002), 293–328.Google Scholar
[9]Li, F. and Liu, M.. Generalized sandwich theorem. K-Theory 1 (1987), 171–184.Google Scholar
[10]Petrov, V.. Overgroups of unitary groups. K-Theory 29 (2003), pp 147–174.Google Scholar
[11]Vaserstein, L. N.. On the normal subgroups of GL n over a ring. Lecture Notes in Math. 854 (1981), pp. 456–465.Google Scholar
[12]Vaserstein, L. N.. The subnormal structure of general linear groups over rings. Math. Proc. Camb. Phil. Soc. 108 (1990), 219–229.Google Scholar
[13]Vavilov, N. A.. Subnormal structure of general linear group. Math. Proc. Camb. Phil. Soc. 107 (1990), 103–196.CrossRefGoogle Scholar
[14]Wilson, J. S.. The normal and subnormal structure of general linear groups. Proc. Camb. Phil. Soc. 71 (1972), 163–177.Google Scholar