Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-26T00:35:34.227Z Has data issue: false hasContentIssue false
  • English
  • Français

Normality in Elementary Subgroups of Chevalley Groups Over Rings

Published online by Cambridge University Press:  20 November 2018

James F. Hurley
James F. Hurley*
Affiliation:
The University of Connecticut, Storrs, Connecticut
Rights & Permissions [Opens in a new window]

Extract

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.

In [6] we have constructed certain normal subgroups G7 of the elementary subgroup GR of the Chevalley group G(L, R) over R corresponding to a finite dimensional simple Lie algebra L over the complex field, where R is a commutative ring with identity. The method employed was to augment somewhat the generators of the elementary subgroup EI of G corresponding to an ideal I of the underlying Chevalley algebra LR;EI is thus the group generated by all xr(t) in G having the property that ter ⊂ I. In [6, § 5] we noted that in general EI actually had to be enlarged for a normal subgroup of GR to be obtained.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 28 , Issue 2 , 01 April 1976 , pp. 420 - 428
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Abe, E., Chevalley groups over local rings, Tôhoku Math. J. 21 (1969), 474–494.Google Scholar
2. Bass, H., Milnor, J. and Serre, J.-P., Solution of the congruence subgroup problem for SLn(n > 3) and Sp2n(n > 2), Inst. Hautes Études Sci. Publ. Math. 33 (1967), 59–137.+3)+and+Sp2n(n+>+2),+Inst.+Hautes+Études+Sci.+Publ.+Math.+33+(1967),+59–137.>Google Scholar
3. Bourbaki, N., Groupes et algèbres de Lie, Chap. IV, V, et VI (Hermann, Paris, 1968).Google Scholar
4. Chevalley, C., Sur certains groupes simples, Tôhoku Math. J. 7 (1955), 14–66.Google Scholar
5. Hurley, J., Ideals in Chevalley algebras, Trans. Amer. Math. Soc. 137 (1969), 245–258.Google Scholar
6. Hurley, J., Some normal subgroups of elementary subgroups of Chevalley groups over rings, Amer. J. Math. 93 (1971), 1059–1069.Google Scholar
7. Klingenberg, W., Linear groups over local rings, Bull. Amer. Math. Soc. 66 (1960), 294–296.Google Scholar
8. Klingenberg, W., Lineare Gruppen ilber lokalen Ringen, Amer. J. Math. 83 (1961), 137–153.Google Scholar
9. Matsumoto, H., Sur les sous-groupes arithmétiques des groupes semi-simples déployés, Ann. Sci. Ecole Norm. Sup. (4) 2 (1969), 1–62.Google Scholar
10. Mennicke, J., Finite factor groups of the unimodular group, Annals of Math. 81 (1965), 31–37.Google Scholar
11. Scott, W., Group theory (Prentice-Hall, Englewood Cliffs, New Jersey, 1964).Google Scholar
12. Stein, M., Generators, relations, and coverings of Chevalley groups over commutative rings, Amer. J. Math. 93 (1971), 965–1004.Google Scholar
13. Steinberg, R., Lectures on Chevalley groups, Yale Univ. Math. Dept., New Haven, Conn., 1967–68.Google Scholar
14. Swan, R., Excision in algebraic K-theory, J. Pure App. Alg. 1 (1971), 221–252.Google Scholar
15. Wilson, J., The normal and subnormal structure of general linear groups, Proc. Cambridge Phil. Soc. 71 (1972), 163–177.Google Scholar