Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-25T01:04:57.756Z Has data issue: false hasContentIssue false
  • English
  • Français

Normality for elementary subgroup functors

Published online by Cambridge University Press:  24 October 2008

Anthony Bak  and
Nikolai Vavilov
Anthony Bak
Affiliation:
Department of Mathematics, University of Bielefeld, Postfach 10 01 31, 33501 Bielefeld, Germany
Nikolai Vavilov
Affiliation:
Department of Mathematics and Mechanics, University of Sanct-Petersburg, Petrodvorets, 198904, Russia
Get access Rights & Permissions [Opens in a new window]

Abstract

We define a notion of group functor G on categories of graded modules, which unifies previous concepts of a group functor G possessing a notion of elementary subfunctor E. We show under a general condition which is easily checked in practice that the elementary subgroup E(M) of G(M) is normal for all quasi-weak Noetherian objects M in the source category of G. This result includes all previous ones on Chevalley and classical groups G of rank ≥ 2 over a commutative or module finite ring M (since such rings are quasi-weak Noetherian) and settles positively unanswered cases of normality for these group functors.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 118 , Issue 1 , July 1995 , pp. 35 - 47
Copyright
Copyright © Cambridge Philosophical Society 1995

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]Abe, E.. Coverings of twisted Chevalley groups over commutative rings. Sci. Repts. Tokyo Kyoiku Daigaku 13 (1977), 194–218.Google Scholar
[2]Abe, E. and Suzuki, K.. On normal subgroups of Chevalley groups over commutative rings. Tôhoku Math. J. 28, N.I (1976), 185–198.Google Scholar
[3]Bak, A.. The stable structure of quadratic modules. Thesis Columbia Univ. (1969).Google Scholar
[4]Bak, A.. On modules with quadratic forms. Lecture Notes Math. 108 (1969), 55–66.Google Scholar
[5]Bak, A.. K-theory of Forms. Ann. Math. Studies 98 (1981) (Princeton Univ. Press).Google Scholar
[6]Bak, A.. Nonabelian K-Theory: The nilpotent class of Kt and general stability. K-Theory 4 (1991), 363–397.CrossRefGoogle Scholar
[7]Bak, A. and Vavilov, N. A.. Structure of hyperbolic unitary groups. I. Elementary subgroup. Submitted.Google Scholar
[8]Bass, H.. K-theory and stable algebra. Publ. Math. Inst. Hautes Et. Sci. 22 (1964), 5–60.Google Scholar
[9]Bass, H.. Unitary algebraic K-theory. Lecture Notes Math. 343 (1973), 57–265.Google Scholar
[10]Cakter, R. W.. Simple groups of Lie type (Wiley, 1972).Google Scholar
[11]Golubchik, I. Z. and Mikhalev, A. V.. Elementary subgroup of a unitary group over a PI-ring. Vestnik Mosk. Univ., ser. 1, Mat., Mekh. N.1 (1985), 30–36.Google Scholar
[12]Hahn, A. J. and O'Meara, O. T.. The classical groups and K-theory (Springer 1989).Google Scholar
[13]Kopeiko, V. I.. The stabilization of symplectic groups over a polynomial ring. Math. U.S.S.R., Sbornik 34 (1978), 655–669.CrossRefGoogle Scholar
[14]Matsumoto, H.. Sur les sous-groupes arithmétiques des groupes semi-simples deployés. Ann. Sci. Ecole Norm. Sup., 4èwie sér. 2 (1969), 1–62.CrossRefGoogle Scholar
[15]Plotkin, E. B.. Parabolic subgroups in 3D 4(R). J. Sov. Math 24, N.4 (1984), 452–457.Google Scholar
[16]Steinberg, R.. Lectures on Chevalley groups (Yale University, 1968).Google Scholar
[17]Suslin, A. A.. On the structure of the general linear group over a polynomial ring. Soviet Math. Izv. 41, N.2 (1977), 503–516.Google Scholar
[18]Suslin, A. A. and Kopeiko, V. A.. Quadratic modules and orthogonal groups over polynomial rings. J. Sov. Math. 20, N.6 (1982), 2665–2691.Google Scholar
[19]Suzuki, K.. On normal subgroups of twisted Chevalley groups over local rings. Sci. Repts. Tokyo Kyoiku Daigaku 13 (1977), 237–249.Google Scholar
[20]Taddei, G.. Invariance du sous-groupe symplectique élémentaire dans le groupe symplectique sur un anneau. C.R. Acad. Sci. Paris, ser. I 295, N.2 (1985), 47–50.Google Scholar
[21]Taddei, G.. Normalité des groupes élémentaire dans les groupes de Chevalley sur un anneau. Contemp. Math. 55, part II (1986), 693–710.Google Scholar
[22]Tulenbaev, M. S.. The Schur multiplier of the group of elementary matrices of finite order. J. Sov. Math. 17, N.4 (1981), 2062–2067.Google Scholar
[23]Vaserstein, L. N.. Stabilization of unitary and orthogonal groups over a ring. Math. U.S.S.R. Sbornik 10 (1970), 307–326.CrossRefGoogle Scholar
[24]Vaserstein, L. N.. On normal subgroups of GLn over a ring. Lecture Notes Math. 854 (1981), 456–465.Google Scholar
[25]Vavilov, N. A.. On subgroups of the split classical groups. Trudy Mat. Inst. Steklov 183, Issue 4 (1991), 27–41.Google Scholar
[26]Vavilov, N. A.. Structure of Chevalley Groups over Commutative Rings. Proc. Conf. Non-Associative Algebras and Related Topics (Hiroshima 1990), World Scientific, Singapore (1991), 219–335.Google Scholar
[27]Vavilov, N. A.. Linear groups over general rings. I, II. In preparation.Google Scholar
[28]Wilson, J. S.. The normal and subnormal structure of general linear groups. Proc. Cambridge Phil. Soc. 71 (1973), 163–177.CrossRefGoogle Scholar