Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-10T14:43:04.220Z Has data issue: false hasContentIssue false
  • English
  • Français

Relative subgroups in Chevalley groups

Published online by Cambridge University Press:  15 March 2010

R. Hazrat ,
V. Petrov  and
N. Vavilov
R. Hazrat
Affiliation:
Department of Pure Mathematics, Queen's University, Belfast BT7 1NN, U.K., r.hazrat@qub.ac.uk
V. Petrov
Affiliation:
Department of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg 198504, Russia, victorapetrov@gmail.com
N. Vavilov
Affiliation:
Department of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg 198504, Russia, nikolai-vavilov@yandex.ru
Get access

Abstract

We finish the proof of the main structure theorems for a Chevalley group G(Φ, R) of rank ≥ 2 over an arbitrary commutative ring R. Namely, we prove that for any admissible pair (A, B) in the sense of Abe, the corresponding relative elementary group E(Φ,R, A, B) and the full congruence subgroup C(Φ, R, A, B) are normal in G(Φ, R) itself, and not just normalised by the elementary group E(Φ, R) and that [E (Φ, R), C(Φ, R, A, B)] = E, (Φ, R, A, B). For the case Φ = F4 these results are new. The proof is new also for other cases, since we explicitly define C (Φ, R, A, B) by congruences in the adjoint representation of G (Φ, R) and give several equivalent characterisations of that group and use these characterisations in our proof.

Keywords

Chevalley groupselementary subgroupcongruence subgroupcommutator formulas
Type
Research Article
Information
Journal of K-Theory , Volume 5 , Issue 3 , June 2010 , pp. 603 - 618
Copyright
Copyright © ISOPP 2010

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

1.Abe, E., Chevalley groups over local rings, Tôhoku Math. J. 21 (1969), no.3, 474–494.CrossRefGoogle Scholar
2.Abe, E., Chevalley groups over commutative rings, Proc. Conf. Radical Theory (Sendai, 1988), Uchida Rokakuho, Tokyo (1989), 1–23.Google Scholar
3.Abe, E., Normal subgroups of Chevalley groups over commutative rings, Algebraic K-Theory and Algebraic Number Theory (Honolulu, HI, 1987), Contemp. Math. 83 (1989), 1–17.Google Scholar
4.Abe, E., Chevalley groups over commutative rings. Normal subgroups and automorphisms, Second International Conference on Algebra (Barnaul, 1991), Contemp. Math. 184 (1995), 13–23.Google Scholar
5.Abe, E., Suzuki, K., On normal subgroups of Chevalley groups over commutative rings, Tôhoku Math. J. 28 (1976), no.1, 185–198.CrossRefGoogle Scholar
6.Bak, A., The stable structure of quadratic modules, Thesis Columbia Univ., 1969.Google Scholar
7.Bak, A., Hazrat, R., Vavilov, N., Localization-completion strikes again: relative K1 is nilpotent by abelian, J. of Pure and Appl. Algebra 213 (2009), 1075–1085CrossRefGoogle Scholar
8.Bak, A., Hazrat, R., Vavilov, N., Structure of hyperbolic unitary groups II. Normal subgroups, To appear.Google Scholar
9.Bak, A., Vavilov, N., Normality for elementary subgroup functors, Math. Proc. Cambridge Philos. Soc. 118 (1995), no. 1, 35–47.CrossRefGoogle Scholar
10.Bak, A., Vavilov, N., Structure of hyperbolic unitary groups I. Elementary subgroups, Algebra Colloq. 7 (2000), no. 2, 159–196.CrossRefGoogle Scholar
11.Bak, A., Vavilov, N., Cubic form parameters, preprint, 2001.Google Scholar
12.Bass, H., Unitary algebraic K-theory, Lecture Notes Math. 343 (1973), 57–265.CrossRefGoogle Scholar
13.Borewicz, Z. I., Vavilov, N. A., The distribution of subgroups in the full linear group over a commutative ring, Proc. Steklov Institute Math. 3 (1985), 27–46.Google Scholar
14.Carter, R. W., Simple groups of Lie type, Wiley, London et al. 1972.Google Scholar
15.Costa, D. L., Keller, G. E., Radix redux: normal subgroups of symplectic groups, J. reine angew. Math. 427 (1992), 51–105.Google Scholar
16.Costa, D. L., Keller, G. E., On the normal subgroups of G(Φ, R)2(A), Trans. Amer. Math. Soc. 351 (1999), no.12, 5051–5088.CrossRefGoogle Scholar
17.Hahn, A. J., O'Meara, O. T., The Classical Groups and K-Theory, Springer, Berlin 1989.CrossRefGoogle Scholar
18.Hazrat, R., Dimension theory and nonstable K1 of quadratic modules, K-Theory 27 (2002), no. 4, 293–328.CrossRefGoogle Scholar
19.Hazrat, R., Vavilov, N., K1 of Chevalley groups are nilpotent, J. of Pure and Appl. Algebra 179 (2003), 99–116.CrossRefGoogle Scholar
20.Hazrat, R., Vavilov, N., Bak's work on the K-theory of rings J. K-Theory, 4 (2009), 1–65.CrossRefGoogle Scholar
21.Hurley, J. F., Ideals in Chevalley algebras, Trans. Amer. Math. Soc. 137 (1969), 245–258.CrossRefGoogle Scholar
22.Hurley, J. F., Some normal subgroups of elementary subgroups of Chevalley groups over rings, Amer. J. Math. 93 (1971), 1059–1069.CrossRefGoogle Scholar
23.Kopeiko, V., The stabilisation of symplectic groups over polynomial rings, Math. USSR Sb. 34 (1978), 655–669.CrossRefGoogle Scholar
24.Fuan, Li, The structure of symplectic group over arbitrary commutative rings, Acta Math. Sinica, New Series 3 (1987), no.3, 247–255.CrossRefGoogle Scholar
25.Fuan, Li, The structure of orthogonal groups over arbitrary commutative rings, Chinese Ann. Math. 10 (1989), no.3, 341–350.Google Scholar
26.Matsumoto, H., Sur les sous-groupes arithmétiques des groupes semi-simples déployés, Ann. Sci. École Norm. Sup. ser. 4, 2 (1969), 1–62.CrossRefGoogle Scholar
27.Petrov, V., Overgroups of unitary groups, K-Theory 29 (2003), no. 3, 147–174.CrossRefGoogle Scholar
28.Petrov, V. A., Odd unitary groups, J. Math. Sci. 130 (2003), no. 3, 4752–4766.CrossRefGoogle Scholar
29.Petrov, V. A., Stavrova, A. K., Elementary subgroups of isotropic reductive groups, St. Petersburg Math. J. 20 (2008), no. 3 (Russian, English translation pending, see also PDMI preprint 1 (2008), 1–20).Google Scholar
30.Stein, M. R., Chevalley groups over commutative rings, Bull. Amer. Math. Soc. 77 (1971), 247–252.CrossRefGoogle Scholar
31.Stein, M. R., Generators, relations and coverings of Chevalley groups over commutative rings, Amer. J. Math. 93 (1971), no.4, 965–1004.CrossRefGoogle Scholar
32.Stein, M. R., Relativizing functors on rings and algebraic K-theory, J. Algebra 19 (1971), no.1, 140–152.CrossRefGoogle Scholar
33.Stepanov, A. V., On the normal structure of the general linear group over a ring, J. Math. Sci. 95 (1999), no.2, 2146–2155.CrossRefGoogle Scholar
34.Stepanov, A., Vavilov, N., Decomposition of transvections: a theme with variations, K-Theory 19 (2000), 109–153.CrossRefGoogle Scholar
35.Stepanov, A. V., Vavilov, N. A., Standard commutator formulae, Vestnik Saint-Petersburg State Univ., ser.1 (2008), no.1, 9–14 (Russian, English translation pending).Google Scholar
36.Suslin, A. A., On the structure of the general linear group over polynomial rings, Soviet Math. Izv. 41 (1977), no. 2503–516.Google Scholar
37.Suslin, A. A., Kopeiko, V. I., Quadratic modules and orthogonal groups over polynomial rings, J. Sov. Math. 20 (1985), no.6, 2665–2691.CrossRefGoogle Scholar
38.Suzuki, K., Normality of the elementary subgroups of twisted Chevalley groups over commutative rings, J. Algebra 175 (1995), no.3, 526–536.CrossRefGoogle Scholar
39.Taddei, G., Normalité des groupes élémentaire dans les groupes de Chevalley sur un anneau, Contemp. Math. 55 II (1986), 693–710.CrossRefGoogle Scholar
40.Vaserstein, L. N., On the normal subgroups of GLn over a ring, Lecture Notes Math. 854 (1981), 456–465.CrossRefGoogle Scholar
41.Vaserstein, L. N., On normal subgroups of Chevalley groups over commutative rings, Tôhoku Math. J. 36 (1986), no.5, 219–230.Google Scholar
42.Vavilov, N., Structure of Chevalley groups over commutative rings, Non-associative algebras and related topics, (Hiroshima, 1990), World Scientific Publishing, London et al., (1991), 219–335.Google Scholar
43.Vavilov, N., An A3-proof of structure theorems for Chevalley groups of types E6 and E7, Int. J. Algebra Comput. 17 (2007), no.5–6, 1283–1298.CrossRefGoogle Scholar
44.Vavilov, N. A., Gavrilovich, M. R., An A2-proof of structure theorems for Chevalley groups of types E6 and E7, St.-Petersburg Math. J. 16 (2005), no.4, 649–672.CrossRefGoogle Scholar
45.Vavilov, N. A., Gavrilovich, M. R., Nikolenko, S. I., Structure of Chevalley groups: the Proof from the Book, J. Math. Sci. 140 (2006), no.5, 626–645.CrossRefGoogle Scholar
46.Vavilov, N. A., Nikolenko, S. I., An A2-proof of structure theorems for Chevalley groups of type F4, St. Petersburg Math. J. 20 (2008), no. 3 (Russian, English translation pending).Google Scholar
47.Vavilov, N. A., Plotkin, E. B., Net subgroups of Chevalley groups, J. Sov. Math. 19 (1982), no.1, 1000–1006.CrossRefGoogle Scholar
48.Vavilov, N. A., Plotkin, E. B., Chevalley groups over commutative rings. I. Elementary calculations, Acta Appl. Math. 45 (1996), 73–115.CrossRefGoogle Scholar
49.Vavilov, N. A., Stavrova, A. K., Basic reductions in the description of normal subgroups, J. Math. Sci. 349 (2007), 30–52 (Russian, English translation pending).Google Scholar