Standardness and standard automorphisms of Chevalley groups, I: the case of rank at least two

Abstract

The author studies the linkage between the standardness and the standard automorphisms of Chevalley groups over rings. It is proved that if H is any standard subgroup of G(R), then each of its automorphisms can be extended to an automorphism of G(R, I), restricted to an automorphism of E(R, I), and an automorphism of E(R, I) can be extended to one of G(R, I). The case of Chevalley groups of rank at least two is treated in this paper. Further results about the case of Chevalley groups of rank one, the case of non-commutative ground ring and some others exceptions will appear elsewhere.

Keywords

Chevalley groups / Standardness / Standard automorphism

References

[1] Abe E.. Chevalley groups over local rings. Tôhoku Math. J., 1969, 21: 474-494

[2] Abe E.. Normal subgroups of Chevalley groups over commutative rings. Cont. Math., 1989, 83: 1-17

[3] Abe E.. Automorphisms of Chevalley groups over commutative rings. St. Petersburg Math. J., 1994, 83(2): 287-300

[4] Abe E.. Chevalley groups over commutative rings, normal subgroups and automorphisms. Contemp. Math., 1995, 184: 13-23

[5] Abe E., Suzuki K.. On normal subgroups of Chevalley groups over commutative rings. Tôhoku Math. J., 1976, 28(4): 185-198

[6] Bardini, C., On automorphisms of congruence subgroups of the Chevalley group of rank one over a local ring, submitted.

[7] Bardini, C., On the automorphisms of Chevalley groups: an A ₂-proof for Al-type, submitted.

[8] Bardini, C., Chevalley groups over rings. The structure problems and beyonds: a survey with compilations, submitted.

[9] Bardini C., Chen Y.. Homomorphisms of two dimensional linear groups over a ring of stable range one. J. Algebra, 2006, 303(1): 30-41

[10] Bass H.. K-theory and stable algebra. Publ. Math. IHES, 1964, 22: 5-60

[11] Borel A., Tits J.. Homomorphismes “abstraits” de groupes algebriques simples. Ann. of Math., 1973, 97: 499-571

[12] Bourbaki N.. Groupes et Algebrés de Lie, Chapters 4–6, 1968, Paris: Hermann

[13] Carter R. W.. Simple Groups of Lie Type, 1972, New York: John Wiley

[14] Chen Y.. Automorphisms of simple Chevalley groups over ℚ-algebras. Tôhoku Math. J., 1995, 47: 81-97

[15] Chen Y.. Isomorphisms of Chevalley groups over algebras. J. Algebra, 2000, 226: 719-741

[16] Costa D. L., Keller G. E.. The E(2,A) sections of SL(2, R). Ann. Math., 1991, 134(2): 159-188

[17] Costa D. L., Keller G. E.. Power residue symbols and the central section of SL(2, A). K-Theory, 1998, 15: 33-92

[18] Costa D. I., Keller G. E.. On the normal subgroups of G ₂(A). Trans. Am. Math. Soc., 1999, 351(12): 5051-5088

[19] Demazure M., Grothendieck A.. Schacute emas en groupes I, 1971, Berlin, New York: Springer-Verlag

[20] Demazure M., Grothendieck A.. Schacute emas en groupes II, 1971, Berlin, New York: Springer-Verlag

[21] Demazure M., Grothendieck A.. Schacute emas en groupes III, 1971, Berlin, New York: Springer-Verlag

[22] Drobotenko V. S., Drobotenko E. S., Ja. Pogorilyak E.. Automorphisms of linear group over commutative rings with minimal condition (in Russian). Upschi Mat. Nauk., 1973, 28(6174): 205-206

[23] Drobotenko V. S., Ya. Pogorilyak E.. Automorphisms of the general linear group over a noncommutative semilocal rings (in Russian). Upschi Mat. Nauk, 1977, 32(3): 157-158

[24] Hahn A. J., James D. G., Weisfeiler B.. Homomorphisms of algebraic and classical groups: a survey. Comm. Math. Soc. Conf. Proceeding, 1984, 4: 249-296

[25] Humphreys J. E.. On the automorphisms of infinite Chevalley groups. Canad. J. Math., 1969, 21: 908-911

[26] James D. G., Waterhouse W., Weisfeiler B.. Abstract homomorphisms of algebraic groups: problems and bibliography. Comm. Alg., 1981, 9(1): 95-114

[27] Mason A. W.. Anomalous normal subgroups of the unimodular group. Comm. Algebra, 1983, 11: 2555-2573

[28] Mason A. W.. Nonstandard, normal subgroups and nonnormal, standard subgroups of the modular group. Canad. Math. Bull., 1989, 32(1): 109-113

[29] Mason A. W., Schweizer A.. Non-standard automorphisms and non-congruence subgroups of SL ₂ over a Dedekind domains contained in function fields. J. Pure and Applied Alg., 2006, 205: 189-209

[30] Merzljakov Y. I.. Automorphisms of two dimensional congruence groups (in Russian). Algebra Logika, 1973, 12: 468-477 English translation in Algebra and Logic, 12(4), 1972, 262–267

[31] Merzlyakov Y. I.. Isomorphisms theory for classical groups from 1976 to 1980 (in Russian), Isomorphisms of Classical Groups over Integral Rings, 1980, Moscow: Mir 252-258

[32] O’Meara O. T., Zassenhaus H.. The automorphisms of the linear congruence groups over Dedekind domains. J. Number Theory, 1969, 1: 211-221

[33] Petechuk V. M.. Homomorphisms of linear groups over rings (in Russian). Mat. Zamet., 1989, 45(2): 83-94 English translation in Math. Notes, 45(1–2), 1989, 144–151

[34] Petechuk V. M.. Homomorphisms of linear groups over commutative rings (in Russian). Mat. Zamet., 1989, 46(5): 50-61

[35] Solazzi R. E.. Isomorphisms theory of congruence groups. Bull. Amer. Math. Soc., 1971, 77: 164-168

[36] Steinberg R.. Automorphisms of finite linear groups. Canad. J. Math., 1960, 12: 606-615

[37] Steinberg R.. Lectures on Chevalley groups, 1968, New Haven, Conn.: Yale University

[38] Suzuki K.. On normal subgroups of twisted Chervalley groups over local rings. Sci. Rep. Tokyo Kyoiku Daigaku Sect. A, 1977, 13: 237-249

[39] Suzuki K.. On the automorphisms of Chevalley groups over p-adic integers rings. Benjamin: Kumamoto J. Ser. (Math.), 1984, 18: 39-47

[40] Taddei G.. Normalité des groupes élémentaire dans les groupes de Chevalley sur un anneau, Applications of Algebraic K-Theory to Algebraic Geometry and Number Theory, Part I, II (Boulder Colo, 1983). Contemp. Math., 1986, Providence, RI: Amer. Mat. Soc. 693-710

[41] Vaserstein L. N.. On normal subgroups of GL _n over a ring, algebraic K-theory-verlay (Proc. Conf. Northwestern Univ., Evanston III, 1980), 1981, Berlin, New York: Springer-Verlag

[42] Vaserstein L. N.. On normal subgroups of Chevalley groups over commutative rings. Tôhoku Math. J., 1986, 38: 219-230

[43] Vaserstein L. N.. Normal subgroups of symplectic groups over rings. K-Theory, 1989, 2: 647-673

[44] Vavilov N. A.. Structure of Chevalley groups over commutative rings. Non-Associative Algebras and Related Topics, Hiroshima 1990, 1991, River Edge, NJ, Singapore: World Sci. Publ. 219-335

[45] Vavilov N. A.. An A ₃-proof of structure theorems for Chevalley groups of type E ₆ and E ₇. International J. of Algebra and Comp., 2007, 17(5–6): 1283-1298

[46] Vavilov N. A., Gavrilovich M. R.. A ₂-proof of structure theorems for Chevalley groups of type E ₆ and E ₇ (in Russian). Algebra Analiz, 2004, 16(4): 54-87

[47] Vavilov N. A., Gavrilovich M. R., Nikolenko S. I.. Structure of Chevalley groups: the proof from the book (in Russian). Zap. Nauch. Sem. POMI, 2006, 330: 36-76

[48] Vavilov, N. A. and Nikolenko, S. I., An A ₂-proof structure theorems for Chevalley groups of type F ₄ and ² E ₆, Algebra Analiz, to appear.

[49] Vavilov N. A., Plotkin E.. Chevalley groups over commutative rings I: elementary calculations. Acta App. Math., 1996, 45: 73-113

[50] Vavilov N. A., Plotkin E., Stepanov A. V.. Computations in Chevalley groups over commutative rings. Dokl. Akad. Nauk. SSSR, 1989, 307(4): 788-791

[51] Wilson J. S.. The normal and subnormal structure of general linear groups. Proc. Camb. Phil. Soc., 1972, 71(2): 163-177