Transvections in the subgroups of the general linear group containing a~non-split maximal torus

V. A. Koibaev

Algebra i analiz, Tome 21 (2009) no. 5, pp. 70-86.

Résumé

The objects of the study are intermediate subgroups of the general linear group $\mathrm{GL}(n,k)$ of degree $n$ over an arbitrary field $k$ that contain a nonsplit maximal torus associated with an extension of degree $n$ of the ground field $k$ (minisotropic torus). It is proved that if an overgroup of a nonsplit torus contains a one-dimensional transformation, then it contains an elementary transvection at some position in every column, and similarly for rows. This result makes it possible to associate net subgroups with groups of the above class and thus forms a base for their further study. This step is motivated by extremely high complexity of the lattice of intermediate subgroups. For a finite field, the lattice of overgroups of a nonsplit maximal torus is essentially determined by subfields intermediate between the ground field and its extension (G. M. Seitz, W. Kantor, R. Dye). Nothing like that holds true for an infinite field; even for the group $\mathrm{GL}(2,k)$ this lattice has much more complicated structure and essentially depends on the arithmetic of the ground field $k$ (Z. I. Borewicz, V. P. Platonov, Chan Ngoc Hoi, the author, and others).

Bibliographie

[1] Bondapenko A. A., “Raspolozhenie podgpupp, sodepzhaschikh nepazvetvlennyi kvadpatichnyi top, v polnoi lineinoi gpuppe stepeni 2 nad lokalnym chislovym polem $(p\neq2)$”, Zap. nauch. semin. POMI, 211, 1994, 67–79 | MR

[2] Bondapenko A. A., “Raspolozhenie podgpupp, sodepzhaschikh nepazvetvlennyi kvadpatichnyi top, v polnoi lineinoi gpuppe stepeni 2 nad lokalnym chislovym polem $(p=2)$”, Zap. nauch. semin. POMI, 211, 1994, 80–90 | MR

[3] Borevich Z. I., Koibaev V. A., “O koltsakh mnozhitelei, svyazannykh s promezhutochnymi podgruppami dlya kvadratichnogo tora”, Vestn. S.-Peterburg. un-ta. Ser. 1, 1993, no. 2, 5–10 | MR | Zbl

[4] Borevich 3. I., Koibaev V. A., Chan Ngok Khoi, “Reshetki podgrupp v $\mathrm{GL}(2,\mathbb Q)$, soderzhaschikh nerasschepimyi tor”, Zap. nauch. semin. LOMI, 191, 1991, 24–43 | Zbl

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

[6] Vavilov N. A., “Podgruppy grupp Shevalle, soderzhaschie maksimalnyi tor”, Tr. Leningr. mat. ob-va, 1, 1990, 64–109 | MR | Zbl

[7] Vavilov N. A., Nesterov V. V., “Geometriya mikrovesovykh torov”, Vladikavkaz. mat. zh., 10:1 (2008), 10–23 | MR

[8] Vavilov N. A., Stepanov A. V., “Nadgruppy poluprostykh grupp”, Vestn. Samar. gos. un-ta. Estestv.-nauch. ser., 2008, no. 3, 51–95 | MR

[9] Dzigoeva V. S., Koibaev V. A., “Promezhutochnye podgruppy v polnoi lineinoi gruppe vtorogo poryadka nad polem ratsionalnykh funktsii, soderzhaschie kvadratichnyi tor”, Vladikavkaz. mat. zh., 10:1 (2008), 27–34 | MR

[10] Dzhusoeva N. A., Koibaev V. A., “Maksimalnye podgruppy, soderzhaschie tor, svyazannye s polem otnoshenii dedekindovoi oblasti”, Zap. nauch. semin. POMI, 289, 2002, 149–153 | MR | Zbl

[11] Koibaev V. A., “Podgruppy gruppy $\mathrm{GL}(2,\mathbb Q)$, soderzhaschie nerasschepimyi maksimalnyi tor”, Dokl. AN SSSR, 312:1 (1990), 36–38 | MR

[12] Koibaev V. A., “Podgruppy gruppy $\mathrm{GL}(2,k)$, soderzhaschie nerasschepimyi maksimalnyi tor”, Zap. nauch. semin. POMI, 211, 1994, 136–145 | MR

[13] Koibaev V. A., “O podgruppakh polnoi lineinoi gruppy, soderzhaschikh maksimalnyi nerasschepimyi tor, svyazannyi s radikalnym rasshireniem”, Vestn. S.-Peterburg. un-ta. Ser. 1, 1995, no. 1, 29–33 | MR

[14] Krupetskii S. L., “O podgruppakh unitarnoi gruppy nad lokalnym polem”, Zap. nauch. semin. LOMI, 94, 1979, 81–103 | MR

[15] Krupetskii S. L., “O podgruppakh unitarnoi gruppy nad diadicheskim lokalnym polem”, Zap. nauch. semin. LOMI, 103, 1980, 79–89 | MR

[16] Chan Ngok Khoi, Raspolozhenie podgrupp v $\mathrm{GL}(2,\mathbb Q)$, soderzhaschikh nerasschepimyi tor, Kand. dis., LGU, L., 1990, 182 pp.

[17] Djoković D. Ž., “Subgroups of compact Lie groups containing a maximal torus are closed”, Proc. Amer. Math. Soc., 83:2 (1981), 431–432 | DOI | MR | Zbl

[18] Dye R. H., “Maximal subgroups of symplectic groups stabilizing spreads. I”, J. Algebra, 87:2 (1984), 493–509 ; “II”, J. London Math. Soc. (2), 40:2 (1989), 215–226 | DOI | MR | Zbl | DOI | MR | Zbl

[19] Dye R. H., “Maximal subgroups of $\mathrm{PS}p_{6n}(q)$ stabilizing spreads of totally isotropic planes”, J. Algebra, 99 (1986), 191–209 | DOI | MR | Zbl

[20] Dye R. H., “Spreads and classes of maximal subgroups of $\mathrm{GL}_n(q)$, $\mathrm{SL}_n(q)$, $\mathrm{PGL}_n(q)$ and $\mathrm{PSL}_n(q)$”, Ann. Mat. Pura Appl. (4), 158 (1991), 33–50 | DOI | MR | Zbl

[21] Kantor W. M., “Linear groups containing a Singer cycle”, J. Algebra, 62:1 (1980), 232–234 | DOI | MR | Zbl

[22] Kariyama K., “On conjugacy classes of maximal tori in classical groups”, J. Algebra, 125:1 (1989), 133–149 | DOI | MR | Zbl

[23] Li Shang Zhi, “Overgroups in $\mathrm{GL}(nr,F)$ of certain subgroups of $\mathrm{SL}(n,K)$”, J. Algebra, 125:1 (1989), 215–235 | DOI | MR | Zbl

[24] Platonov V. P., “Subgroups of algebraic groups over a local or global field containing a maximal torus”, C. R. Acad. Sci. Paris Sér. I Math., 318:10 (1994), 899–903 | MR | Zbl

[25] Seitz G. M., “Subgroups of finite groups of Lie type”, J. Algebra, 61:1 (1979), 16–27 | DOI | MR | Zbl

[26] Seitz G. M., “The root subgroups for maximal tori in finite groups of Lie type”, Pacific J. Math., 106:1 (1983), 153–244 | MR | Zbl

[27] Vavilov N., “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