Abstract
In the present paper, it is proved that if R is a commutative semilocal ring all the residue fields of which contain at least 3n + 2 elements, then for every subgroup H of the special linear group SL(n, R), n ≥ 3, containing the diagonal subgroup SD(n, R) there exists a unique D-net σ of ideals of R such that Γ(σ)≤H≤NΓ(σ). In works by Z. I. Borewicz and the author, similar results were established for GL n over semilocal rings and for SL n over fields. Later I. Hamdan obtained a similar description for the very special case of uniserial rings. Bibliography: 76 titles.
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References
Z. I. Borewicz, “On parabolic subgroups in linear groups over a semilocal ring,” Vestn. LGU, Ser. 1, No. 13, 16–24 (1976).
Z. I. Borewicz, “On parabolic subgroups in the special linear group over a semilocal ring,” Vestn. LGU, Ser. 1, No. 19, 29–34 (1976).
Z. I. Borewicz, “A description of subgroups of the general linear group that contain the group of diagonal matrices,” Zap. Nauchn. Semin. LOMI, 64, 12–29 (1976).
Z. I. Borewicz and N. A. Vavilov, “The subgroups of the general linear group over a semilocal ring that contain the group of diagonal matrices,” Tr. Steklov Mat. Inst. AN SSSR, 148, 43–57 (1978).
Z. I. Borewicz and N. A. Vavilov, “On the definition of a net subgroup,” Zap. Nauchn. Semin. LOMI, 132, 26–33 (1983).
Z. I. Borewicz and J. O. Lesama-Serrano, “The group of invertible elements of a semiperfect ring,” in: Rings and Modules. Limit Theorems in Probability Theory, Vol. 1, Leningrad (1986), pp. 14–67.
Bui Xuan Hai, “Arrangement of subgroup in the special linear group over a skew field with an infinite center,” Zap. Nauchn. Semin. LOMI, 175, 5–11 (1989).
Bui Xuan Hai, “The subgroups of the special linear group over a skew field that contain the group of diagonal matrices,” Zap. Nauchn. Semin. LOMI, 211, 91–103 (1994).
N. A. Vavilov, “On the subgroups of the general linear group over a semilocal ring that contain the group of diagonal matrices,” Vestn. LGU, No. 1, 10–15 (1981).
N. A. Vavilov, “The Bruhat decomposition for subgroups that contain the group of diagonal matrices. I, II,” Zap. Nauchn. Semin. LOMI, 103, 20–30 (1980); 114, 50–61 (1982).
N. A. Vavilov, “On subgroups of the special linear group that contain the group of diagonal matrices. I–V,” Vestn. LGU, Ser. I, No. 4, 3–7 (1985); No. 2, 10–15 (1986); No. 2, 3–8 (1987); No. 3, 10–15 (1988); No. 2, 10–15 (1993).
N. A. Vavilov, “The Bruhat decomposition of one-dimensional transformations,” Vestn. LGU, Ser. I, No. 3, 14–20 (1986).
N. A. Vavilov, Subgroups of split classical groups, Doctoral Thesis, Leningrad State University (1987).
N. A. Vavilov, “On subgroups of split orthogonal groups. I, II,” Sib. Mat. Zh., 29,No. 3, 12–25 (1988); Zap. Nauchn. Semin. POMI, 265, 42–63 (1999).
N. A. Vavilov, “Subgroups of split classical groups,” Tr. Steklov Mat. Inst. AN SSSR, 183, 29–41 (1990).
N. A. Vavilov, “Subgroups of Chevalley groups that contain a maximal torus,” Tr. Leningr. Mat. Obshch., 1, 64–109 (1990).
N. A. Vavilov, “On subgroups of the spinor group that contain a split maximal torus. I, II,” Zap. Nauchn. Semin. LOMI, 191, 49–75 (1991); 289, 37–56 (2002).
N. A. Vavilov, “Unipotent elements in the subgroups of extend Chevalley groups that contain a split maximal torus,” Dokl. RAN, 328,No. 5, 536–539 (1993).
N. A. Vavilov and E. V. Dybkova, “Subgroups of general symplectic group that contain the group of diagonal matrices. I, II,” Zap. Nauchn. Semin. LOMI, 103, 31–47 (1980); 132, 44–56 (1983).
N. A. Vavilov and M. Yu. Mitrofanov, “Intersections of two Bruhat cells,” Dokl. RAN, 377,No. 1, 7–10 (2001).
N. A. Vavilov and A. A. Semenov, “Long root semisimple elements in Chevalley groups,” Dokl. RAN, 338,No. 6, 725–727 (1994).
N. A. Vavilov and V. A. Petrov, “On overgroups of Ep(2l, R),” Algebra Analiz, 15,No. 4, 72–114 (2003).
N. A. Vavilov and I. Khamdan, “On subgroups of the general linear group over a local field,” Izv. Vuzov, Matematika, 12, 8–15 (1989).
W. Hołubowski, “Subgroups of an isotropic orthogonal group that contain the centralizer of a maximal torus,” Zap. Nauchn. Semin. POMI, 191, 76–79 (1991).
W. Hołubowski, “The structure of isotropic orthogonal groups,” Ph.D. Thesis, St.Petersburg State University (1991).
E. V. Dybkova, “On net subgroups of hyperbolic unitary groups,” Algebra Analiz, 9,No. 4, 87–96 (1997).
E. V. Dybkova, “Overdiagonal subgroups of a hyperbolic unitary group for a good form ring over a field,” Zap. Nauchn. Semin. POMI, 236, 87–96 (1997).
E. V. Dybkova, “On the conjugacy of net subgroups in a hyperbolic unitary group over a field,” Vestn. SPbGU, Ser. 1, No. 1, 8–11 (1997).
E. V. Dybkova, “Form nets and the lattice of overdiagonal subgroups in the symplectic group over a field of characteristic 2,” Algebra Analiz, 10,No. 4, 113–129 (1998).
E. V. Dybkova, “On overdiagonal subgroups of a hyperbolic unitary group over a noncommutative skew field,” Zap. Nauchn. Semin. POMI, 289, 154–206 (2002).
E. V. Dybkova, xxx“Overdiagonal subgroups of a hyperbolic unitary group for a good form ring over a noncommutative skew field,” Zap. Nauchn. Semin. POMI, 305, 121–135 (2003).
E. V. Dybkova, “The Borewicz theorem for a hyperbolic unitary group over a noncommutative skew field,” Zap. Nauchn. Semin. POMI, 321, 136–167 (2005).
E. V. Dybkova, “Subgroups of hyperbolic unitary groups,” Doctoral Thesis, St.Petersburg State Univ. (2006).
A. E. Egorov and A. A. Panin, “The conjugacy theorem for subgroups of SLn that contain the group of diagonal matrices,” Zap. Nauchn. Semin. POMI, 272, 177–185 (2000).
A. E. Zalesskii, “Linear groups,” Usp. Mat. Nauk, 36,No. 6, 57–107 (1981).
A. E. Zalesskii, “Linear groups,” in: Scientific Reviews. Algebra, Topology, Geometry, VINITI, Moscow (1983), pp. 135–182.
A. E. Zalesskii, “Linear groups,” in: Scientific Reviews. Fundamental Directions. Algebra, VINITI, Moscow (1989), pp. 114–228.
V. A. Koibaev, “Subgroups in GL(2, K) that contain a nonsplit maximal torus,” Zap. Nauchn. Semin. POMI, 211, 136–145 (1994).
A. S. Kondratiev, “Subgroups of finite Chevalley groups,” Usp. Mat. Nauk, 41,No. 1, 57–96 (1986).
K. Yu. Lavrov, “Subgroups of the general linear group over a local field,” Zap. Nauchn. Semin. POMI, 272, 242–258 (2000).
K. Yu. Lavrov, “Subgroups of orthogonal groups of even order over a local field,” Zap. Nauchn. Semin. POMI, 321, 240–250 (2005).
M. Yu. Mitrofanov, “Matroids describing Bruhat cells,” Zap. Nauchn. Semin. POMI, 319, 244–260 (2004).
A. A. Panin, “Galois theory for a class of complete Dedekind structures,” Zap. Nauchn. Semin. POMI, 236, 129–132 (1997).
A. A. Panin and A. V. Yakovlev, “Galois theory for a class of Dedekind structures,” Zap. Nauchn. Semin. POMI, 236, 133–148 (1997).
V. A. Petrov, “Odd unitary groups,” Zap. Nauchn. Semin. POMI, 305, 195–225 (2003).
A. Z. Simonyan, “Galois theory for modular lattices,” Ph.D. Thesis, St.Petersburg State University (1992).
E. A. Sopkina, “A classification of group subschemes of GLn that contain a split maximal torus,” Zap. Nauchn. Semin. POMI, 321, 281–296 (2005).
E. A. Sopkina, “Group subschemes of reductive groups,” Ph.D. Thesis, St.Petersburg State University (2006).
E. A. Filippova, “Subgroups of the spinor group that contain a split maximal torus. III,” Zap. Nauchn. Semin. POMI, 289, 287–299 (2002).
I. Khamdan, “Subgroups of the general linear group over a local field,” Ph.D. Thesis, Leningrad State University (1987).
I. Khamdan, “Subgroups of the special linear group of a local ring of principal ideals,” in: Rings and Linear Groups, Krasnodar (1988), pp. 119–126.
A. Bak and N. Vavilov, “Structure of hyperbolic unitary groups. I. Elementary subgroups,” Algebra Colloq., 7,No. 2, 159–196 (2000).
Bui Xuan Hai, “On subgroups in the special linear group over a division algebra that contain a subgroup of diagonal matrices,” J. Pure Appl. Algebra, 121,No. 1, 53–67 (1997).
A. M. Cohen, H. Cuypers, and H. Sterk, “Linear groups generated by reflection tori,” Canad. J. Math., 51,No. 6, 1149–1174 (1999).
D. Costa and G. Keller, “Radix redux: normal subgroups of symplectic group,” J. Reine Angew. Math., 427, 51–105 (1992).
A. J. Hahn and O. T. O’Meara, The Classical Groups and K-Theory, Springer, Berlin et al. (1989).
A. L. Harebov and N. A. Vavilov, “On the lattice of subgroups of Chevalley groups containing a split maximal torus,” Comm. Algebra, 24,No. 1, 109–133 (1996).
O. H. King, “On subgroups of the special linear group containing the special orthogonal group,” J. Algebra, 96,No. 1, 178–193 (1985).
O. H. King, “On subgroups of the special linear group containing the diagonal subgroup,” J. Algebra, 132,No. 1, 198–204 (1990).
O. H. King, “The subgroup structure of classical groups,” Contemp. Math., 131,No. 1, 209–215 (1992).
O. H. King, “Subgroups of the special linear group, containing the groups of diagonal matrices,” Preprint Univ. Newcastle u.T. (2005).
Li Shang Zhi, “The maximality of monomial subgroups of linear groups over division rings,” J. Algebra, 127,No. 1, 22–39 (1989).
M. Yu. Mitrofanov and N. A. Vavilov, “Overgroups of the diagonal subgroup via small Bruhat cells,” Algebra Colloq. (to appear).
V. A. Petrov, “Overgroups of unitary groups,” K-Theory, 29, 77–108 (2003).
V. P. Platonov, “Subgroups of an algebraic group over a local or global field containing a maximal torus,” C. R. Acad. Sci. Paris, Sér. Math., 318,No. 10, 899–903 (1994).
G. M. Seitz, “Subgroups of finite groups of Lie type,” J. Algebra, 61,No. 1, 16–27 (1979).
G. M. Seitz, “On the subgroup structure of classical groups,” Commun. Algebra, 10,No. 8, 875–885 (1982).
G. M. Seitz, “Root subgroups for maximal tori in finite groups of Lie type,” Pacif. J. Math., 106,No. 1, 153–244 (1983).
E. A. Sopkina, “Classification of all connected subgroup schemes of a reductive group containing a split maximal torus,” Preprint POMI 05/2006 (2006).
E. A. Sopkina, “Subgroup schemes of a reductive group containing a split maximal torus,” Preprint POMI 10/2006 (2006).
N. A. Vavilov, “Subgroups of split orthogonal groups in even dimensions,” Bull. Acad. Polon. Sci., Ser. Sci. Math., 29,No. 9–10, 425–429 (1981).
N. A. Vavilov, “A conjugacy theorem for subgroups of GLn containing the group of diagonal matrices,” Colloq. Math., 54,No. 1, 9–14 (1987).
N. A. Vavilov, “Intermediate subgroups in Chevalley groups,” in: Proceedings of the Conference on Groups of Lie Type and Their Geometries (Como — 1993), Cambridge Univ. Press (1995), pp. 233–280.
N. A. Vavilov, “Unipotent elements in subgroups which contain a split maximal torus,” J. Algebra, 176, 356–367 (1995).
N. A. Vavilov, “Geometry of 1-tori in GLn,” Preprint Univ. Bielefeld, No. 8 (1995).
N. A. Vavilov, “Subgroups of SLn over a semilocal ring,” Preprint Univ. Bielefeld, No. 111 (1998).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 343, 2007, pp. 33–53.
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Vavilov, N.A. Subgroups of SLn over a semilocal ring. J Math Sci 147, 6995–7004 (2007). https://doi.org/10.1007/s10958-007-0525-3
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DOI: https://doi.org/10.1007/s10958-007-0525-3