St. Petersburg Mathematical Society

Author(s): N. A. Vavilov; V. A. Petrov
Translated by: the authors
Original publication: Algebra i Analiz, tom 15 (2003), vypusk 4.
Journal: St. Petersburg Math. J. 15 (2004), 515-543.
MSC (2000): Primary 20G35
Posted: July 6, 2004
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Abstract: Let

be a commutative ring, and let $l\ge 2$ ; for

it is assumed additionally that

has no residue fields of two elements. The subgroups of the general linear group $\operatorname{GL}(n,R)$ that contain the elementary symplectic group $\operatorname{Ep}(2l,R)$ are described. In the case where

is a field, similar results were obtained earlier by Dye, King, and Shang Zhi Li.

1.: H. Bass, Algebraic -theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491 (40:2736)
2.: H. Bass, J. Milnor, and J.-P. Serre, Solution of the congruence subgroup problem for $\operatorname{SL}_n$ ( $n\ge 3$ ) and $\operatorname{Sp}_{2n}$ ( $n\ge 2$ ), Inst. Hautes Études Sci. Publ. Math. No. 33 (1967), 59-137. MR 0244257 (39:5574)
3.: E. L. Bashkirov, Linear groups containing the special unitary group of non-zero index, Vestsi Akad. Navuk BSSR Ser. Fiz.-Mat. Navuk 1986, no. 5, 120-121 (complete text of manuscript sits at VINITI 7.08.1985, no. 5897-85 Dep.). (Russian)
4.: -, Linear groups containing the symplectic group, Vestsi Akad. Navuk BSSR Ser. Fiz.-Mat. Navuk 1987, no. 3, 116-117 (complete text of manuscript sits at VINITI 11.04.1986, no. 2616-B86 Dep.). (Russian)
5.: -, Linear groups that contain the group $\operatorname{Sp}_{n}(K)$ over a field of characteristic 2, Vestsi Akad. Navuk BSSR Ser. Fiz.-Mat. Navuk 1991, no. 4, 21-26. (Russian) MR 1141150 (92j:20041)
6.: -, Linear groups that contain the commutant of an orthogonal group of index greater than 1, Sibirsk. Mat. Zh. 33 (1992), no. 5, 15-21; English transl., Siberian Math. J. 33 (1992), no. 5, 754-759 (1993). MR 1197069 (94e:20061)
7.: -, On subgroups of the general linear group over the skew field of quaternions containing the special unitary group, Sibirsk. Mat. Zh. 39 (1998), no. 6, 1251-1266; English transl., Siberian Math. J. 39 (1998), no. 6, 1080-1092. MR 1672621 (99m:20125)
8.: Z. I. Borevich and N. A. Vavilov, Arrangement of subgroups in the general linear group over a commutative ring, Trudy Mat. Inst. Steklov. 165 (1984), 24-42; English transl. in Proc. Steklov Inst. Math. 1985, no. 3. MR 0752930 (86e:20052)
9.: N. A. Vavilov, Subgroups of split classical groups, Doctor's Thesis, Leningrad. Gos. Univ., Leningrad, 1987. (Russian). MR 0953017 (89g:20074)
10.: -, The structure of split classical groups over a commutative ring, Dokl. Akad. Nauk SSSR 299 (1988), no. 6, 1300-1303; English transl., Soviet Math. Dokl. 37 (1988), no. 2, 550-553. MR 0947412 (89j:20053)
11.: -, Subgroups of split orthogonal groups over a ring, Sibirsk. Mat. Zh. 29 (1988), no. 4, 31-43; English transl., Siberian Math. J. 29 (1988), no. 4, 537-547 (1989). MR 0969101 (90c:20061)
12.: -, Subgroups of splittable classical groups, Trudy Mat. Inst. Steklov. 183 (1990), 29-42; English transl., Proc. Steklov Inst. Math. 1991, no. 4, 27-41. MR 1092012 (92d:20065)
13.: -, On subgroups of the general symplectic group over a commutative ring, Rings and Modules. Limit Theorems of Probability Theory, No. 3, S.-Peterburg. Gos. Univ., St. Petersburg, 1993, pp. 16-38. (Russian) MR 1351048 (96j:20066)
14.: -, Subgroups of split orthogonal groups over a commutative ring, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 281 (2001), 35-59. (Russian) MR 1875717 (2002j:20099)
15.: N. A. Vavilov and E. V. Dybkova, Subgroups of the general symplectic group containing the group of diagonal matrices, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 103 (1980), 31-47; English transl. in J. Soviet Math. 24 (1984), no. 4. MR 0618492 (82h:20054)
16.: N. A. Vavilov and V. A. Petrov, On subgroups of $\operatorname{EO}(2l,R)$ , Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 272 (2000), 68-85; English transl., J. Math. Sci. 116 (2003), no. 1, 2917-2925. MR 1811793 (2001m:20078)
17.: L. N. Vasershtein, On the stabilization of the general linear group over a ring, Mat. Sb. (N. S.) 79 (1969), no. 3, 405-424; English transl., Math. USSR-Sb. 8 (1969), 383-400. MR 0267009 (42:1911)
18.: -, Stabilization of unitary and orthogonal groups over a ring with involution, Mat. Sb. (N. S.) 81 (1970), no. 3, 328-351; English transl. in Math. USSR-Sb. 10 (1970). MR 0269722 (42:4617)
19.: L. N. Vasershtein and A. A. Suslin, Serre's problem on projective modules over polynomial rings, and algebraic -theory, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 5, 993-1054; English transl. in Math. USSR-Izv. 10 (1976), no. 5. MR 0447245 (56:5560)
20.: I. Z. Golubchik, Normal subgroups of the orthogonal group over the associative ring with involution, Uspekhi Mat. Nauk 30 (1975), no. 6, 165. (Russian)
21.: -, Subgroups of the general linear $\operatorname{GL}_n(R)$ over an associative ring , Uspekhi Mat. Nauk 39 (1984), no. 1, 125-126; English transl. in Russian Math. Surveys 39 (1984), no. 1. MR 0733962 (85j:20042)
22.: -, Normal subgroups of the linear and unitary groups over associative rings, Spaces over Algebras, and Some Problems in the Theory of Nets, Bashkir. Gos. Ped. Inst., Ufa, 1985, pp. 122-142. (Russian) MR 0975035
23.: V. I. Kopeiko, Stabilization of symplectic groups over a ring of polynomials, Mat. Sb. (N. S.) 106 (1978), no. 1, 94-107; English transl., Math. USSR-Sb. 34 (1978), no. 5, 655-669. MR 0497932 (80f:13008)
24.: O. T. O'Meara, Lectures on symplectic groups, Univ. Notre Dame, 1976.
25.: R. Steinberg, Lectures on Chevalley groups, Yale Univ., New Haven, Conn., 1968. MR 0466335 (57:6215)
26.: A. V. Stepanov, Stability conditions in the theory of linear groups over rings, Ph.D. Thesis, Leningrad. Gos. Univ., Leningrad, 1987. (Russian)
27.: -, On the distribution of subgroups normalized by a given subgroup, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 198 (1991), 92-102; English transl., J. Soviet Math. 64 (1993), no. 1, 769-776. MR 1164862 (93g:20047)
28.: A. A. Suslin, The structure of the special linear group over rings of polynomials, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 2, 235-252; English transl. in Math. USSR. Izv. 11 (1977), no. 2 (1978). MR 0472792 (57:12482)
29.: E. Abe, Whitehead groups of Chevalley groups over polynomial rings, Comm. Algebra 11 (1983), no. 12, 1271-1307. MR 0697617 (85d:20038)
30.: -, Chevalley groups over commutative rings, Radical Theory (Sendai, 1988), Uchida Rokakuho, Tokyo, 1989, pp. 1-23. MR 0999577 (91a:20047)
31.: -, Normal subgroups of Chevalley groups over commutative rings, Algebraic -Theory and Algebraic Number Theory (Honolulu, HI, 1987), Contemp. Math., vol. 83, Amer. Math. Soc., Providence, RI, 1989. MR 0991973 (91a:20046)
32.: E. Abe and K. Suzuki, On normal subgroups of Chevalley groups over commutative rings, Tôhoku Math. J. (2) 28 (1976), no. 2, 185-198. MR 0439947 (55:12828)
33.: M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent. Math. 76 (1984), no. 3, 469-514. MR 0746539 (86a:20054)
34.: A. Bak, The stable structure of quadratic modules, Thesis, Columbia Univ., 1969.
35.: -, -theory of forms, Ann. of Math. Stud., vol. 98, Princeton Univ. Press, Princeton, NJ, 1981. MR 0632404 (84m:10012)
36.: -, Nonabelian K-theory: the nilpotent class of K and general stability, K-Theory 4 (1991), 363-397. MR 1115826 (92g:19001)
37.: A. Bak and A. Stepanov, Dimension theory and nonstable K-theory for net groups, Rend. Sem. Mat. Univ. Padova 106 (2001), 207-253. MR 1876221 (2002j:18012)
38.: A. Bak and N. Vavilov, Normality for elementary subgroup functors, Math. Proc. Cambridge Philos. Soc. 118 (1995), no. 1, 35-47. MR 1329456 (96d:20046)
39.: -, Structure of hyperbolic unitary groups. . Elementary subgroups, Algebra Colloq. 7 (2000), no. 2, 159-196. MR 1810843 (2002b:20070)
40.: -, Structure of hyperbolic unitary groups. . Normal subgroups (to appear).
41.: H. Bass, Unitary algebraic -theory, : Hermitian -theory and geometric applications (Proc. Conf. Seattle Res. Center, Battelle Memorial Inst., 1972), Lecture Notes in Math., vol. 343, Springer, Berlin, 1973, pp. 57-265. MR 0371994 (51:8211)
42.: R. Carter, Simple groups of Lie type, Pure App. Math., vol. 28, Wiley, London etc., 1972. MR 0407163 (53:10946)
43.: D. Costa and G. Keller, The sections of $\operatorname{SL}(2,A)$ , Ann. of Math. (2) 134 (1991), no. 1, 159-188. MR 1114610 (92f:20047)
44.: -, Radix redux: normal subgroups of symplectic groups, J. Reine Angew. Math. 427 (1992), no. 1, 51-105. MR 1162432 (93h:20053)
45.: L. Di Martino and N. A. Vavilov, -generation of $\operatorname{SL}(n,q)$ . . Cases , Comm. Algebra 22 (1994), no. 4, 1321-1347. MR 1261262 (95f:20076)
46.: R. H. Dye, Interrelations of symplectic and orthogonal groups in characteristic two, J. Algebra 59 (1979), no. 1, 202-221. MR 0541675 (81c:20028)
47.: -, On the maximality of the orthogonal groups in the symplectic groups in characteristic two, Math. Z. 172 (1980), no. 3, 203-212. MR 0581439 (81h:20060)
48.: -, Maximal subgroups of $\operatorname{GL}_{2n}(K)$ , $\operatorname{SL}_{2n}(K)$ , $\operatorname{PGL}_{2n}(K)$ , and $\operatorname{PSL}_{2n}(K)$ associated with symplectic polarities, J. Algebra 66 (1980), no. 1, 1-11. MR 0591244 (81j:20061)
49.: F. Grünewald, J. Mennicke, and L. N. Vaserstein, On symplectic groups over polynomial rings, Math. Z. 206 (1991), no. 1, 35-56. MR 1086811 (92a:19001)
50.: A. Hahn and O. T. O'Meara, The classical groups and K-theory, Grundlehren Math. Wiss., vol. 291, Springer-Verlag, Berlin-New York, 1989. MR 1007302 (90i:20002)
51.: R. Hazrat, Dimension theory and nonstable of quadratic modules, -Theory 27 (2002), 293-328. MR 1962906 (2004a:19005)
52.: R. Hazrat and N. A. Vavilov, K of Chevalley groups are nilpotent, J. Pure Appl. Algebra 179 (2003), 99-116. MR 1958377
53.: W. Jehne, Die Struktur der symplektischen Gruppe über lokalen und dedekindschen Ringen, Sitzungsber. Heidelb. Akad. Wiss. Math.-Natur. Kl. 1962/64, 187-235. MR 0175999 (31:275)
54.: O. H. King, On subgroups of the special linear group containing the special orthogonal group, J. Algebra 96 (1985), no. 1, 178-193. MR 0808847 (87b:20057)
55.: -, On subgroups of the special linear group containing the special unitary group, Geom. Dedicata 19 (1985), no. 3, 297-310. MR 0815209 (87c:20081)
56.: -, Subgroups of the special linear group containing the diagonal subgroup, J. Algebra 132 (1990), no. 1, 198-204. MR 1060843 (91c:20056)
57.: F. Kirchheimer, Die Normalteiler der symplektischen Gruppen über beliebigen lokalen Ringen, J. Algebra 50 (1978), no. 1, 228-241. MR 0578492 (58:28209)
58.: P. Kleidman and M. Liebeck, The subgroup structure of the finite classical groups, London Math. Soc. Lecture Note Ser., vol. 129, Cambridge Univ. Press, Cambridge, 1990. MR 1057341 (91g:20001)
59.: W. Klingenberg, Symplectic groups over local rings, Amer. J. Math. 85 (1963), 232-240. MR 0153749 (27:3710)
60.: M. Knus, A. S. Merkurjev, M. Rost, and J. P. Tignol, The book of involutions, Amer. Math. Soc. Colloq. Publ., vol. 44, Amer. Math. Soc., Providence, RI, 1998. MR 1632779 (2000a:16031)
61.: Fu An Li, The structure of symplectic groups over arbitrary commutative rings, Acta Math. Sinica (N.S.) 3 (1987), no. 3, 247-255. MR 0916269 (88m:20098)
62.: -, The structure of orthogonal groups over arbitrary commutative rings, Chinese Ann. Math. Ser. B 10 (1989), no. 3, 341-350. MR 1027673 (90k:20084)
63.: Shang Zhi Li, Overgroups of $\operatorname{SU}(n,K,f)$ or $\Omega(n,K,f)$ in $\operatorname{GL}(n,K)$ , Geom. Dedicata 33 (1990), no. 3, 241-250. MR 1050412 (91g:11038)
64.: -, Overgroups of a unitary group in $\operatorname{GL}(2,K)$ , J. Algebra 149 (1992), no. 2, 275-286. MR 1172429 (93e:20067)
65.: -, Overgroups in $\operatorname{GL}(n,F)$ of a classical group over a subfield of , Algebra Colloq. 1 (1994), no. 4, 335-346. MR 1301157 (95i:20068)
66.: Shang Zhi Li and Zong Li Wei, Overgroups of a symplectic group in a linear group over a Euclidean ring, J. Univ. Sci. Technol. China 32 (2002), no. 2, 127-134. (Chinese) MR 1911131 (2003d:20071)
67.: M. Newman, Matrix completion theorems, Proc. Amer. Math. Soc. 94 (1985), no. 1, 39-45. MR 0781052 (86d:15009)
68.: V. A. Petrov, On the first cohomology of unitary Steinberg groups (to appear). (English)
69.: -, Overgroups of unitary groups (to appear). (English) MR 2028500
70.: M. R. Stein, Generators, relations and coverings of Chevalley groups over commutative rings, Amer. J. Math. 93 (1971), no. 3, 965-1004. MR 0322073 (48:437)
71.: -, Stability theorems for $\operatorname{K}_1$ , $\operatorname{K}_2$ and related functors modeled on Chevalley groups, Japan. J. Math. (N.S.) 4 (1978), no. 1, 77-108. MR 0528869 (81c:20031)
72.: A. Stepanov, Non-standard subgroups between and $\operatorname{GL}(n,A)$ , Algebra Colloq. (to appear).
73.: A. Stepanov and N. Vavilov, Decomposition of transvections: a theme with variations, -Theory 19 (2000), 109-153. MR 1740757 (2000m:20076)
74.: G. Taddei, Invariance du sous-groupe symplectique élémentaire dans le groupe symplectique sur un anneau, C. R. Acad. Sci Paris Sér I Math. 295 (1982), no. 2, 47-50. MR 0676359 (84c:20058)
75.: -, Normalité des groupes élémentaires dans les groupes de Chevalley sur un anneau, Applications of Algebraic -Theory to Algebraic Geometry and Number Theory, Part I, II (Boulder, Colo., 1983), Contemp. Math., vol. 55, Amer. Math. Soc., Providence, RI, 1986, pp. 693-710. MR 0862660 (88a:20054)
76.: F. Timmesfeld, Abstract root subgroups and quadratic action. With an appendix by A. E. Zalesskii, Adv. Math. 142 (1999), no. 1, 1-150. MR 1671440 (2000g:20051)
77.: J. Tits, Systèmes générateurs de groupes de congruences, C. R. Acad. Sci. Paris Sér A-B 283 (1976), A693-A695. MR 0424966 (54:12924)
78.: L. N. Vaserstein, On the normal subgroups of the $\operatorname{GL}_n$ over a ring, Algebraic -Theory, Evanston 1980 (Proc. Conf., Northwestern Univ., Evanston, Ill., 1980), Lecture Notes in Math., vol. 854, Springer, Berlin-New York, 1981, pp. 454-465. MR 0618316 (83c:20058)
79.: -, An answer to a question of M. Newman on matrix completion, Proc. Amer. Math. Soc. 97 (1986), no. 2, 189-196. MR 0835863 (88e:18012)
80.: -, On normal subgroups of Chevalley groups over commutative rings, Tôhoku Math. J. 38 (1986), no. 2, 219-230. MR 0843808 (87k:20081)
81.: -, Normal subgroups of orthogonal groups over commutative rings, Amer. J. Math. 110 (1988), no. 5, 955-973. MR 0961501 (89i:20071)
82.: -, Normal subgroups of symplectic groups over rings, Proceedings of Research Symposium on -Theory and its Applications (Ibadan, 1987), -Theory 2 (1989), no. 5, 647-673. MR 0999398 (90f:20064)
83.: L. N. Vaserstein and Hong You, Normal subgroups of classical groups over rings, J. Pure Appl. Algebra 105 (1995), no. 1, 93-106. MR 1364152 (96k:20096)
84.: N. A. Vavilov, Structure of Chevalley groups over commutative rings, Nonassociative Algebras and Related Topics (Hiroshima, 1990), World Sci. Publishing, River Edge, NJ, 1991, pp. 219-335. MR 1150262 (92k:20090)
85.: -, Intermediate subgroups in Chevalley groups, Groups of Lie Type and their Geometries (Como, 1993), London Math. Soc. Lecture Note Ser., vol. 207, Cambridge Univ. Press, Cambridge, 1995, pp. 233-280. MR 1320525 (96c:20085)
86.: -, The work of Borevich on linear groups, and beyond, Proc. Internat. Algebra Conf. (St. Petersburg, 2002), Marcel Dekker, 2003 (to appear).
87.: J. S. Wilson, The normal and subnormal structure of general linear groups, Proc. Cambridge Philos. Soc. 71 (1972), 163-177. MR 0291304 (45:398)
88.: Hong You and Baodong Zheng, Overgroups of symplectic group in linear group over local rings, Comm. Algebra 29 (2001), no. 6, 2313-2318. MR 1845112 (2002d:20083)
89.: E. V. Dybkova, Overdiagonal subgroups of the hyperbolic unitary group for a good form ring over a field, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 236 (1997), 87-96; English transl., J. Math. Sci. 95 (1999), no. 2, 2096-2101. MR 1754447 (2001k:20101)
90.: -, On overdiagonal subgroups of the hyperbolic unitary group for a skew field, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 289 (2002), 154-206. (Russian) MR 1949740 (2004a:20056)

N. A. Vavilov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskii Prospekt 28, St. Petersburg, 198504, Russia

V. A. Petrov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskii Prospekt 28, St. Petersburg 198504, Russia

DOI: 10.1090/S1061-0022-04-00820-9
PII: S 1061-0022(04)00820-9
Received by editor(s): 18/FEB/2003
Posted: July 6, 2004
Additional Notes: The present paper has been written in the framework of the RFBR projects nos. 01-01-00924 and 00-01-00441, and INTAS 00-566. The theorem on decomposition of unipotents mentioned in \S13 is a part of first author's joint work with A. Bak and was carried out at the University of Bielefeld with the support of AvH-Stiftung, SFB-343, and INTAS 93-436. At the final stage, the work of the authors was supported by express grants of the Russian Ministry of Higher Education `Geometry of root subgroups' PD02-1.1-371 and `Overgroups of semisimple groups' E02-1.0-61.
Copyright of article: Copyright 2004, American Mathematical Society

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