Overgroups of elementary symplectic groups
Algebra i analiz, Tome 15 (2003) no. 4, pp. 72-114.

Voir la notice de l'article provenant de la source Math-Net.Ru

@article{AA_2003_15_4_a2,
     author = {N. A. Vavilov and V. A. Petrov},
     title = {Overgroups of elementary symplectic groups},
     journal = {Algebra i analiz},
     pages = {72--114},
     publisher = {mathdoc},
     volume = {15},
     number = {4},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AA_2003_15_4_a2/}
}
TY  - JOUR
AU  - N. A. Vavilov
AU  - V. A. Petrov
TI  - Overgroups of elementary symplectic groups
JO  - Algebra i analiz
PY  - 2003
SP  - 72
EP  - 114
VL  - 15
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AA_2003_15_4_a2/
LA  - ru
ID  - AA_2003_15_4_a2
ER  - 
%0 Journal Article
%A N. A. Vavilov
%A V. A. Petrov
%T Overgroups of elementary symplectic groups
%J Algebra i analiz
%D 2003
%P 72-114
%V 15
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AA_2003_15_4_a2/
%G ru
%F AA_2003_15_4_a2
N. A. Vavilov; V. A. Petrov. Overgroups of elementary symplectic groups. Algebra i analiz, Tome 15 (2003) no. 4, pp. 72-114. http://geodesic.mathdoc.fr/item/AA_2003_15_4_a2/

[1] Bass X., Algebraicheskaya $K$-teoriya, Mir, M., 1973 | MR | Zbl

[2] Bass X., Milnor Dzh., Serr Zh.-P., “Reshenie kongruentsproblemy, dlya $SL_n$ ($n\ge3$) i $Sp_{2n}$”, Matematika. Period. sb. per. in. st., 14:6 (1970), 64–128 ; 15:1 (1971), 44–60 | Zbl | Zbl

[3] Bashkirov E. L., “O lineinykh gruppakh, soderzhaschikh spetsialnuyu unitarnuyu gruppu nenulevogo indeksa”, Izv. AN BSSR. Ser. fiz.-mat. nauk, 1986, no. 5, 120–121; деп. в ВИНИТИ 7 авг. 1985 г., No 5897-85

[4] Bashkirov E. L., “O lineinykh gruppakh, soderzhaschikh simplekticheskuyu gruppu”, Izv. AN BSSR. Ser. fiz.-mat. nauk, 1987, no. 3, 116–117; деп. в ВИНИТИ 11 апр. 1986 г. No 2616-В86

[5] Bashkirov E. L., “O lineinykh gruppakh, soderzhaschikh gruppu $Sp_n(K)$ nad polem kharakteristiki 2”, Izv. AN BSSR. Ser. fiz.-mat. nauk, 1991, no. 4, 21–26 | MR | Zbl

[6] Bashkirov E. L., “O lineinykh gruppakh, soderzhaschikh kommutant ortogonalnoi gruppy indeksa bolshe 1”, Sib. mat. zh., 33:5 (1992), 15–21 | MR

[7] Bashkirov E. L., “O podgruppakh polnoi lineinoi gruppy nad telom kvaternionov, soderzhaschikh spetsialnuyu unitarnuyu gruppu”, Sib. mat. zh., 39:6 (1998), 1251–1266 | MR | Zbl

[8] Borevich Z. I., Vavilov N. A., “Raspolozhenie podgrupp v polnoi lineinoi gruppe nad kommutativnym koltsom”, Tr. Mat. in-ta AN SSSR, 165, 1984, 24–42 | MR

[9] Vavilov N. A., Podgruppy rasschepimykh klassicheskikh grupp, Dokt. diss., LGU, L., 1987, 334 pp.

[10] Vavilov N. A., “Stroenie rasschepimykh klassicheskikh grupp nad kommutativnym koltsom”, Dokl. AN SSSR, 299:6 (1988), 1300–1303 | MR | Zbl

[11] Vavilov N. A., “O podgruppakh rasschepimykh ortogonalnykh grupp nad koltsom”, Sib. mat. zh., 29:4 (1988), 31–43 | MR | Zbl

[12] Vavilov N. A., “O podgruppakh rasschepimykh klassicheskikh grupp”, Tr. Mat. in-ta AN SSSR, 183, 1990, 29–42 | MR

[13] Vavilov N. A., “O podgruppakh polnoi simplekticheskoi gruppy nad kommutativnym koltsom.”, Koltsa i moduli. Predelnye teoremy teorii veroyatnostei, vyp. 3, SPbGU, SPb., 1993, 16–38 | MR

[14] Vavilov N. A., “Podgruppy rasschepimykh ortogonalnykh grupp nad kommutativnym koltsom”, Zap. nauch. semin. POMI, 281, 2001, 35–59 | Zbl

[15] Vavilov N. A., Dybkova E. V., “Podgruppy polnoi simplekticheskoi gruppy, soderzhaschie gruppu diagonalnykh matrits”, Zap. nauch. semin. LOMI, 103, 1980, 31–47 | MR | Zbl

[16] Vavilov N. A., Petrov V. A., “O nadgruppakh $\mathrm{EO}(2l,R)$”, Zap. nauch. semin. POMI, 272, 2000, 68–85 | MR | Zbl

[17] Vasershtein L. N., “O stabilizatsii obschei lineinoi gruppy nad koltsom”, Mat. sb., 79(121):3(7) (1969), 405–424 | MR

[18] Vasershtein L. N., “Stabilizatsiya unitarnykh i ortogonalnykh grupp nad koltsom s involyutsiei”, Mat. sb., 81(123):3 (1970), 328–351 | MR

[19] Vasershtein L. N., Suslin A. A., “Problema Serra o proektivnykh modulyakh nad koltsami mnogochlenov i algebraicheskaya $K$-teoriya”, Izv. AN SSSR. Ser. mat., 40:5 (1976), 993–1054 | MR

[20] Golubchik I. Z., “O normalnykh delitelyakh ortogonalnoi gruppy nad assotsiativnym koltsom s involyutsiei”, Uspekhi mat. nauk, 30:6(186) (1975), 165–165 | Zbl

[21] Golubchik I. Z., “O podgruppakh polnoi lineinoi gruppy $\mathrm{GL}_n(R)$ nad assotsiativnym koltsom $R$”, Uspekhi mat. nauk, 39:1 (1984), 125–126 | MR | Zbl

[22] Golubchik I. Z., “O normalnykh delitelyakh lineinoi i unitarnoi grupp nad assotsiativnymi koltsami”, Prostranstva nad algebrami i nekotorye voprosy teorii setei,, BGPI, Ufa, 1985, 122–142 | MR

[23] Kopeiko V. I., “Stabilizatsiya simplekticheskikh grupp nad koltsom mnogochlenov”, Mat. sb., 106(148):1(5) (1978), 94–107 | MR | Zbl

[24] O'Mira O. T., Lektsii o simplektichestkh gruppakh, Mir, M., 1979

[25] Steinberg R., Lektsii o gruppakh Shevalle, Mir, M., 1975 | MR | Zbl

[26] Stepanov A. V., Usloviya stabilnosti v teorii lineinykh grupp nad koltsami, Kand. diss., LGU, L., 1987, 1–112

[27] Stepanov A. V., “O raspolozhenii podgrupp, normalizuemykh fiksirovannoi”, Zap. nauch. semin. LOMI, 198, 1991, 92–102 | Zbl

[28] Suslin A. A., “O strukture spetsialnoi lineinoi gruppy nad koltsami mnogochlenov”, Izv. AN SSSR. Ser. mat., 41:2 (1977), 235–252 | MR | Zbl

[29] Abe E., “Whitehead groups of Chevalley groups over polynomial rings”, Comm. Algebra, 11:12 (1983), 1271–1307 | DOI | MR | Zbl

[30] Abe E., “Chevalley groups over commutative rings”, Radical Theory (Sendai, 1988), Uchida Rokakuho, Tokyo, 1989, 1–23 | MR

[31] Abe E., “Normal subgroups of Chevalley groups over commutative rings”, Algebraic $K$-Theory and Algebraic Number Theory (Honolulu, HI, 1987), Contemp. Math., 83, Amer. Math. Soc., Providence, RI, 1989, 1–17 | MR

[32] Abe E., Suzuki K., “On normal subgroups of Chevalley groups over commutative rings”, Tohoku Math. J. (2), 28:2 (1976), 185–198 | DOI | MR | Zbl

[33] Aschbacher M., “On the maximal subgroups of the finite classical groups”, Invent. Math., 76:3 (1984), 469–514 | DOI | MR | Zbl

[34] Bak A., The stable structure of quadratic modules, Thesis, Columbia Univ., 1969

[35] Bak A., $K$-theory of forms, Ann. of Math. Stud., 98, Princeton Univ. Press, Princeton, NJ, 1981 | MR | Zbl

[36] Bak A., “Nonabelian $K$-theory: the nupotent class of $K_1$ and general stability”, $K$-Theory, 4 (1991), 363–397 | DOI | MR | Zbl

[37] Bak A., Stepanov A., “Dimension theory and nonstable $K$-theory for net groups”, Rend. Sem. Mat. Univ. Padova, 106 (2001), 207–253 | MR | Zbl

[38] Bak A., Vavilov N., “Normality for elementary subgroup functors”, Math. Proc. Cambridge Philos. Soc., 118:1 (1995), 35–47 | DOI | MR | Zbl

[39] Bak A., Vavilov N., “Structure of liyperbolic unitary groups. I. Elementary subgroups”, Algebra Colloq., 7:2 (2000), 159–196 | DOI | MR | Zbl

[40] Bak A., Vavilov N., Structure of hyperbolic unitary groups. II. Normal subgroups, v pechati

[41] Bass H., “Unitary algebraic $K$-theory. III: Hermitian $K$-theory and geometric applications”, Proc. Conf. Seattle Res. Center (Battelle Memorial Inst., 1972), Lecture Notes in Math., 343, Springer, Berlin, 1973, 57–265 | MR

[42] Carter R., Simple groups of lie type, Pure Appl. Math., 28, Wiley, London etc., 1972 | MR

[43] Costa D., Keller G., “The $E(2,A)$ sections of $\mathrm{SL}(2,A)$”, Ann. of Math. (2), 134:1 (1991), 159–188 | DOI | MR | Zbl

[44] Costa D., Keller G., “Radix redux: normal subgroups of symplectic groups”, J. Reine Angew. Math., 427:1 (1992), 51–105 | MR | Zbl

[45] Di Martino L., Vavilov N. A., “$(2,3)$-generation of $\mathrm{SL}(n,q)$. I. Cases $n=5,6,7$”, Comm. Algebra, 22:4 (1994), 1321–1347 | DOI | MR | Zbl

[46] Dye R. H., “Interrelations of symplectic and orthogonal groups in characteristic two”, J. Algebra, 59:1 (1979), 202–221 | DOI | MR | Zbl

[47] Dye R. H., “On the maximality of the orthogonal groups in the symplectic groups in characteristic two”, Math. Z., 172:3 (1980), 203–212 | DOI | MR | Zbl

[48] Dye R. H., “Maximal subgroups of $\mathrm{GL}_{2n}(K)$, $\mathrm{SL}_{2n}(K)$, $\mathrm{PGL}_{2n}(K)$, and $\mathrm{PSL}_{2n}(K)$ associated with symplectic polarities”, J. Algebra, 66:1 (1980), 1–11 | DOI | MR | Zbl

[49] Grünewald F., Mennicke J., Vaserstein L. N., “On symplectic groups over polynomial rings”, Math. Z., 206:1 (1991), 35–56 | DOI | MR

[50] Hahn A., O'Meara O. T., The classical groups and $K$-theory, Grundlehren Math. Wiss., 291, Springer-Verlag, Berlin–New York, 1989 | MR | Zbl

[51] Hazrat R., “Dimension theory and nonstable $K_1$ of quadratic modules”, $K$-Theory, 27 (2002), 293–328 | DOI | MR | Zbl

[52] Hazrat R., Vavilov N. A., “$K_1$ of Chevalley groups are nilpotent”, J. Pure Appl. Algebra, 179 (2003), 99–116 | DOI | MR | Zbl

[53] Jehne W., “Die Struktur der symplektischen Gruppe über lokalen und dedekindschen Ringen”, Sitzungsber. Heidelb. Akad. Wiss. Math.-Natur. Kl., 3 (1962/64), 187–235 | MR

[54] King O. H., “On subgroups of the special linear group containing tlte special orthogonal group”, J. Algebra, 96:1 (1985), 178–193 | DOI | MR | Zbl

[55] King O. H., “On subgroups of the special linear group containing the special unitary group”, Geom. Dedicate, 19:3 (1985), 297–310 | MR | Zbl

[56] King O. H., “Subgroups of the special linear group containing the diagonal subgroup”, J. Algebra, 132:1 (1990), 198–204 | DOI | MR | Zbl

[57] Kirchheimer F., “Die Normalteiler der symplektischen Gruppen über beliebigen lokalen Ringen”, J. Algebra, 50:1 (1978), 228–241 | DOI | MR | Zbl

[58] Kleidman P., Liebeck M., The subgroup structure of the finite classical groups, London Math. Soc. Lecture Note Ser., 129, Cambridge Univ. Press, Cambridge, 1990 | MR | Zbl

[59] Klingenberg W., “Symplectic groups over local rings”, Amer. J. Math., 85 (1963), 232–240 | DOI | MR | Zbl

[60] Knus M., Merkurjev A. S., Rost M., Tignol J.-P., The book of involutions, Amer. Math. Soc. Colloq. Publ., 44, Amer. Math. Soc., Providence, RI, 1998 | MR | Zbl

[61] Li Fu An., “The structure of symplectic groups over arbitrary commutative rings”, Acta Math. Sinica (N.S.), 3:3 (1987), 247–255 | MR

[62] Li Fu An., “The structure of orthogonal groups over arbitrary commutative rings”, Chinese Ann. Math. Ser. B, 10:3 (1989), 341–350 | MR

[63] Li Shang Zhi., “Overgroups of $SU(n,K,f)$ or $\Omega(n,K,f)$ in $\mathrm{GL}(n,K)$”, Geom. Dedicate, 33:3 (1990), 241–250 | MR | Zbl

[64] Li Shang Zhi., “Overgroups of a unitary group in $\mathrm{GL}(2,K)$”, J. Algebra, 149:2 (1992), 275–286 | DOI | MR | Zbl

[65] Li Shang Zhi., “Overgroups in $\mathrm{GL}(n,F)$ of a classical group over a subfleld of $F$F”, Algebra Colloq., 1:4 (1994), 335–346 | MR | Zbl

[66] Li Shang Zhi., Wei Zong Li, “Overgroups of a symplectic group in a linear group over a Euclidean ring”, J. Univ. Sci. Technol. China, 32:2 (2002), 127–134 (Chinese) | MR | Zbl

[67] Newman M., “Matrix completion theorems”, Proc. Amer. Math. Soc., 94:1 (1985), 39–45 | DOI | MR | Zbl

[68] Petrov V. A., On the first cohomology of unitary Steinberg groups, v pechati

[69] Petrov V. A., “Overgroups of unitary groups”, $K$-Theory, 29:3 (2003), 147–174 | DOI | MR | Zbl

[70] Stein M. R., “Generators, relations and coverings of Chevalley groups over commutative rings”, Amer. J. Math., 93:3 (1971), 965–1004 | DOI | MR | Zbl

[71] Stein M. R., “Stability theorems for $K_1$, $K_2$ and related functors modeled on Chevalley groups”, Japan J. Math. (N.S.), 4:1 (1978), 77–108 | MR | Zbl

[72] Stepanov A., “Non-standard subgroups between $E_n(R)$ and $\mathrm{GL}_n(A)$”, Algebra Colloq., 11:3 (2004), 321–334 | MR | Zbl

[73] Stepanov A., Vavilov N., “Decomposition of transvections: a theme with variations”, $K$-Theory, 19 (2000), 109–153 | DOI | MR | Zbl

[74] Taddei G., “Invariance du sous-groupe symplectique élémentaire dans le groupe symplectique sur un anneau”, C. R. Acad. Sci Paris Sér I Math., 295:2 (1982), 47–50 | MR | Zbl

[75] Taddei G., “Normalité des groupes élémentaires 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., 55, Amer. Math. Soc., Providence, RI, 1986, 693–710 | MR

[76] Timmesfeld F., “Abstract root subgroups and quadratic action”, With an appendix by A. E. Zalesskii, Adv. Math., 142:1 (1999), 1–150 | DOI | MR | Zbl

[77] Tits J., “Systèmes générateurs de groupes de congruences”, C. R. Acad. Sci. Paris Sér A-B, 283 (1976), A693–A695 | MR

[78] Vaserstein L. N., “On the normal subgroups of the $\mathrm{GL}_n$ over a ring”, Algebraic $K$-Theory Evanston 1980, Proc. Conf. (Northwestern Univ., Evanston, Ill., 1980), Lecture Notes in Math., 854, Springer, Berlin–New York, 1981, 454–465 | MR

[79] Vaserstein L. N., “An answer to a question of M. Newman on matrix completion”, Proc. Amer. Math. Soc., 97:2 (1986), 189–196 | DOI | MR | Zbl

[80] Vaserstein L. N., “On normal subgroups of Chevalley groups over commutative rings”, Tohoku Math. J., 38:2 (1986), 219–230 | DOI | MR | Zbl

[81] Vaserstein L. N., “Normal subgroups of orthogonal groups over commutative rings”, Amer. J. Math., 110:5 (1988), 955–973 | DOI | MR | Zbl

[82] Vaserstein L. N., “Normal subgroups of symplectic groups over rings”, $K$-Theory, 2:5 (1989), 647–673, Proceedings of Research Symposium on $K$-Theory and its Applications (Ibadan, 1987) | DOI | MR | Zbl

[83] Vaserstein L. N., You Hong, “Normal subgroups of classical groups over rings”, J. Pure Appl. Algebra, 105:1 (1995), 93–106 | DOI | MR

[84] Vavilov N. A., “Structure of Chevalley groups over commutative rings”, Nonassociative Algebras and Related Topics (Hiroshima, 1990), World Sci. Publishing, River Edge, NJ,, 1991, 219–335 | MR | Zbl

[85] Vavilov N. A., “Intermediate subgroups in Chevalley groups”, Groups of Lie Type and their Geometries (Como, 1993), London Math. Soc. Lecture Note Ser., 207, Cambridge Univ. Press, Cambridge, 1995, 233–280 | MR | Zbl

[86] Vavilov N. A., “The work of Borewicz on linear groups, and beyond”, Proc. Internet. Algebra Conf. (St-Petersburg, 2002), Marcel Dekker, 2003 (to appear)

[87] Wilson J. S., “The normal and subnormal structure of general linear groups”, Proc. Cambridge Philos. Soc., 71 (1972), 163–177 | DOI | MR | Zbl

[88] You Hong, Zheng Baodong, “Overgroups of symplectic group in linear group over local rings”, Comm. Algebra, 29:6 (2001), 2313–2318 | DOI | MR | Zbl

[89] Dybkova E. V., “Naddiagonalnye podgruppy giperbolicheskoi unitarnoi gruppy dlya khoroshego formennogo koltsa nad polem”, Zap. nauch. semin. POMI, 236, 1997, 87–96 | MR | Zbl

[90] Dybkova E. V., “O naddiagonalnykh podgruppakh giperbolicheskoi unitarnoi gruppy nad nekommutativnym telom”, Zap. nauch. semin. POMI, 289, 2002, 154–206 | MR | Zbl