Geometry of 1-tori in $\mathrm{GL}(n,T)$

N. Vavilov

Algebra i analiz, Tome 19 (2007) no. 3, pp. 119-150.

Résumé

We describe the orbits of the general linear group $\mathrm{GL}(n,T)$ over a skew field $T$ acting by simultaneous conjugation on pairs of 1-tori, i.e., subgroups conjugate to $\operatorname{diag}(T^*,1,\dots, 1)$ and identify the corresponding spans. We also provide some applications of these results to the description of intermediate subgroups and generation. These results were partly superseded by A. Cohen, H. Cuypers, and H. Sterk, but our proofs use only elementary matrix techniques. As another application of our methods, we enumerate the orbits of $\mathrm{GL}(n,T)$ on pairs of a 1-torus and a root subgroup, and identify the corresponding spans. This paper constitutes an elementary invitation to a series of much more technical works by the author and V. Nesterov, where similar results are established for microweight tori in Chevalley groups over a field.

Détail

Keywords: General linear group, unipotent root subgroups, pseudoreflections, one-dimensional tori, diagonal subgroup, orbitals.

Bibliographie

[1] Bashkirov E. L., “O lineinykh gruppakh, porozhdennykh dvumya dlinnymi kornevymi podgruppami”, Sib. matem. zhurn., 34:2 (1993), 15–22 | MR | Zbl

[2] Bashkirov E. L., “O podgruppakh spetsialnoi lineinoi gruppy stepeni 2 nad beskonechnym polem”, Matem. sb., 187:2 (1996), 19–36 | MR | Zbl

[3] Bashkirov E. L., “Lineinye gruppy, soderzhaschie kornevuyu podgruppu”, Sib. matem. zhurn., 37:6 (1996), 1238–1255 | MR | Zbl

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

[5] Bashkirov E. L., “O podgruppakh polnoi lineinoi gruppy stepeni 4 nad telom kvaternionov, soderzhaschikh spetsialnuyu unitarnuyu gruppu indeksa 1”, Algebra i analiz, 13:3 (2001), 18–42 | MR

[6] Bashkirov E. L., “Gruppa $\mathrm{Spin}_8$ i nekotorye podgruppy unitarnoi gruppy stepeni 4 nad telom kvaternionov”, Algebra i analiz, 13:3 (2001), 43–64 | MR

[7] Bashkirov E. L., Lineinye gruppy nad telami, soderzhaschie podgruppy kvadratichnykh unipotentnykh elementov, Dokt. dis., Belorusskii gos. un-t Informatiki i Radioelektroniki, 2006, 1–270

[8] Borevich Z. I., “Opisanie podgrupp polnoi lineinoi gruppy, soderzhaschikh gruppu diagonalnykh matrits”, Zap. nauch. semin. LOMI, 64, 1976, 12–29 | Zbl

[9] Borevich Z. I., Vavilov N. A., “Podgruppy polnoi lineinoi gruppy nad polulokalnym koltsom, soderzhaschie gruppu diagonalnykh matrits”, Tr. MIAN, 148, 1978, 43–57 | MR | Zbl

[10] Borevich Z. I., Koibaev V. A., “Podgruppy polnoi lineinoi gruppy nad polem iz pyati elementov”, Algebra i teoriya chisel, 3, Kabard.-Balkar. un-t, Ordzhonikidze (Vladikavkaz), 1978, 9–32

[11] Borel A., “Svoistva i lineinye predstavleniya grupp Shevalle”, Seminar po algebraicheskim gruppam, Mir, M., 1973, 9–59 | MR

[12] Borel A., Tits Zh., “Reduktivnye gruppy”, Matematika, Period. sb. perev. in. statei, 11, no. 1, 1967, 43–111; No 2, 3–31

[13] Vavilov N. A., “O podgruppakh polnoi lineinoi gruppy nad polulokalnym koltsom, soderzhaschikh gruppu diagonalnykh matrits”, Vestn. Leningr. un-ta. Ser. 1, 1981, no. 1, 10–15 | MR | Zbl

[14] Vavilov N. A., “Razlozhenie Bryua dlya podgrupp, soderzhaschikh gruppu diagonalnykh matrits, I, II”, Zap. nauch. semin. LOMI, 103, 1980, 20–30 ; 114, 1982, 50–61 | MR | Zbl | MR | Zbl

[15] Vavilov N. A., “O podgruppakh spetsialnoi lineinoi gruppy, soderzhaschikh gruppu diagonalnykh matrits, I–IV”, Vestn. Leningr. un-ta. Ser. I, 1985, no. 4, 3–7 ; 1986, No 1, 10–15 ; 1987, No 2, 3–8 ; 1988, No 3, 10–15 | MR | Zbl | MR | Zbl | MR | Zbl | MR | Zbl

[16] Vavilov N. A., “Razlozhenie Bryua odnomernykh preobrazovanii”, Vestn. Leningr. un-ta. Ser. I, 1986, no. 3, 14–20 | MR | Zbl

[17] Vavilov N. A., “O geometrii dlinnykh kornevykh podgrupp v gruppakh Shevalle”, Vestn. Leningr. un-ta. Ser. I, 1988, no. 1, 8–11 | MR | Zbl

[18] Vavilov N. A., “Vesovye elementy grupp Shevalle”, Dokl. AN SSSR, 298:3 (1988), 524–527 | MR | Zbl

[19] Vavilov N. A., “Teoremy sopryazhennosti dlya podgrupp rasshirennykh grupp Shevalle, soderzhaschikh rasschepimyi maksimalnyi tor”, Dokl. AN SSSR, 299:2 (1988), 269–272 | MR | Zbl

[20] Vavilov N. A., “Razlozhenie Bryua dlinnykh kornevykh poluprostykh elementov v gruppakh Shevalle”, Koltsa i moduli. Predelnye teoremy teorii veroyatnostei, vyp. 2, LGU, L., 1988, 18–39 | MR

[21] Vavilov N. A., “O vzaimnom raspolozhenii dlinnoi i korotkoi kornevykh podgrupp v gruppe Shevalle”, Vestn. Leningr. un-ta. Ser. I, 1989, no. 1, 3–7 | MR | Zbl

[22] Vavilov N. A., “Razlozhenie Bryua dvumernykh preobrazovanii”, Vestn. Leningr. un-ta. Ser. I, 1989, no. 3, 3–7 | MR | Zbl

[23] Vavilov N. A., “Lineinye gruppy, porozhdennye odnoparametricheskimi gruppami odnomernykh preobrazovanii”, UMN, 44:1(265) (1989), 189–190 | MR | Zbl

[24] Vavilov N. A., “Kornevye poluprostye elementy i troiki kornevykh unipotentnykh podgrupp v gruppakh Shevalle”, Voprosy algebry, vyp. 4, Gomelsk. un-t, Universitetskoe, Minsk, 1989, 162–173 | MR

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

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

[27] Vavilov N. A., “Unipotentnye elementy v podgruppakh rasshirennykh grupp Shevalle, soderzhaschikh rasschepimyi maksimalnyi tor”, Dokl. RAN, 328:5, (1993), 536–539 | MR | Zbl

[28] Vavilov N. A., Dybkova E. V., “Podgruppy polnoi simplekticheskoi gruppy, soderzhaschie gruppu diagonalnykh matrits, I, II”, Zap. nauch. semin. LOMI, 103, 1980, 31–47 ; 132, 1983, 44–56 | MR | Zbl | MR

[29] Vavilov N. A., “Podgruppy gruppy $\mathrm{SL}_n$ nad polulokalnym koltsom”, Zap. nauch. semin. POMI, 343, 2007, 33–53 | MR

[30] Vavilov N. A., Mitrofanov M. Yu., “Peresecheniya dvukh kletok Bryua”, Dokl. RAN, 377:1 (2001), 7–10 | MR | Zbl

[31] Vavilov N. A., Semenov A. A., “Razlozhenie Bryua dlinnykh kornevykh torov v gruppakh Shevalle”, Zap. nauch. semin. LOMI, 175, 1989, 12–23 | Zbl

[32] Vavilov N. A., Semenov A. A., “Dlinnye kornevye poluprostye elementy v gruppakh Shevalle”, Dokl. RAN, 338:6 (1994), 725–727 | MR | Zbl

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

[34] Dybkova E. V., “Naddiagonalnye podgruppy giperbolicheskoi unitarnoi gruppy dlya khoroshego formennogo koltsa nad nekommutativnym telom”, Zap. nauch. semin. POMI, 305, 2003, 121–135 | MR

[35] Dybkova E. V., “Teorema Borevicha dlya giperbolicheskoi unitarnoi gruppy nad nekommutativnym telom”, Zap. nauch. semin. POMI, 321, 2005, 136–167 | Zbl

[36] Dybkova E. V., Podgruppy giperbolicheskikh unitarnykh grupp, Dokt. dis., SPb gos. un-t, 2006

[37] Zalesskii A. E., “Lineinye gruppy”, UMN, 36:5(221) (1981), 57–107 | MR | Zbl

[38] Zalesskii A. E., “Lineinye gruppy”, Itogi nauki i tekhn. Algebra. Topologiya. Geometriya, 21, VINITI, M., 1983, 135–182 | MR

[39] Zalesskii A. E., “Lineinye gruppy”, Algebra 4, Itogi nauki i tekhn. Sovrem. probl. mat. Fundam. napravleniya, 37, VINITI, M., 1989, 114–234 | MR | Zbl

[40] Zalesskii A. E., Serezhkin V. N., “Lineinye gruppy, porozhdennye transvektsiyami”, Izv. AN SSSR. Ser. mat., 40:1 (1976), 26–49 | MR | Zbl

[41] Zalesskii A. E., Serezhkin V. N., “Lineinye gruppy, porozhdennye psevdootrazheniyami”, Vestsi AN BSSR. Ser. fiz.-mat. navuk, 1977, no. 5, 9–16 | MR | Zbl

[42] Zalesskii A. E., Serezhkin V. N., “Konechnye lineinye gruppy, porozhdennye otrazheniyami”, Izv. AN SSSR. Ser. mat., 44:6 (1980), 1279–1307 | MR | Zbl

[43] Kon P. M., Svobodnye koltsa i ikh svyazi, Mir, M., 1975 | MR

[44] Koibaev V. A., “Primery nemonomialnykh lineinykh grupp bez transvektsii”, Zap. nauch. semin. LOMI, 71, 1977, 153–154 | MR | Zbl

[45] Koibaev V. A., “Podgruppy polnoi lineinoi gruppy nad polem iz chetyrekh elementov”, Algebra i teoriya chisel, vyp. 4, Kabard.-Balkar. un-t, Nalchik, 1979, 21–31

[46] Koibaev V. A., “Opisanie $D$-polnykh podgrupp v polnoi lineinoi gruppe nad polem iz trekh elementov”, Zap. nauch. semin. LOMI, 103, 1980, 76–78 | MR | Zbl

[47] Koibaev V. A., “Podgruppy polnoi lineinoi gruppy nad polem iz trekh elementov”, Strukturnye svoistva algebraicheskikh sistem, Kabard.-Balkar. gos. un-t, Nalchik, 1981, 56–68

[48] Kondratev A. S., “Podgruppy konechnykh grupp Shevalle”, UMN, 41:1(247) (1986), 57–96 | MR

[49] Korlyukov A. V., “Lineinye gruppy, porozhdennye dvumernymi elementami poryadka $r\ge 5$”, Vestn. Mosk. gos. un-ta, 1983, no. 5, 19–22 | MR | Zbl

[50] Korlyukov A. V., “Konechnye lineinye gruppy nad polem nulevoi kharakteristiki, porozhdennye dvumernymi elementami poryadka 3 i 4”, Arifmeticheskoe i podgruppovoe stroenie konechnykh grupp, Nauka i tekhn., Minsk, 1986, 75–86 | MR

[51] Korlyukov A. V., “Konechnye lineinye gruppy, porozhdennye kvadratichnymi elementami poryadka 4”, Vestsi AN BSSR. Ser. fiz.-mat. navuk, 1986, no. 4, 38–40 | MR | Zbl

[52] Nesterov V. V., “Pary korotkikh kornevykh podgrupp v gruppakh Shevalle”, Dokl. RAN, 357:3 (1997), 302–305 | MR | Zbl

[53] Nesterov V. V., “Raspolozhenie dlinnoi i korotkoi kornevykh podgrupp v gruppe Shevalle tipa $G_2$”, Zap. nauch. semin. POMI, 272, 2000, 273–285 | MR | Zbl

[54] Nesterov V. V., “Pary korotkikh kornevykh podgrupp v gruppe Shevalle tipa $G_2$”, Zap. nauch. semin. POMI, 281, 2001, 253–273 | MR | Zbl

[55] Nesterov V. V., “Porozhdenie par korotkikh kornevykh podgrupp v gruppakh Shevalle”, Algebra i analiz, 16:6 (2004), 172–208 | MR

[56] Petrov V. A., “Nechetnye unitarnye gruppy”, Zap. nauch. semin. POMI, 305, 2003, 195–225

[57] Semenov A. A., “Razlozhenie Bryua kornevykh poluprostykh podgrupp v spetsialnoi lineinoi gruppe”, Zap. nauch. semin. LOMI, 160, 1987, 239–246 | Zbl

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

[59] Shevalle K., “O nekotorykh prostykh gruppakh”, Matematika, Period sb. perev. in. statei, 2, no. 1, 1958, 3–58

[60] Aschbacher M., Seitz G. M., “Involutions in Chevalley groups over fields of even order”, Nagoya Math. J., 63 (1976), 1–91 | MR | Zbl

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

[62] Bashkirov E. L., “Some completely reducible linear groups over a quaternion division ring containing a root subgroup”, Comm. Algebra, 31:12 (2003), 5727–5754 | DOI | MR | Zbl

[63] Bashkirov E. L., “Irreducible linear groups of degree 3 over a quaternion division ring containing a root subgroup”, Comm. Algebra, 32:5 (2004), 1747–1761 | DOI | MR | Zbl

[64] Bashkirov E. L., “Irreducible linear groups of degree 4 over a quaternion division algebra that contain a subgroup $(T_3(K,\Phi_0)1)$”, J. Algebra, 287:2 (2005), 319–350 | DOI | MR | Zbl

[65] Baumann B., Ho C. Y., “Linear groups generated by a pair of quadratic action subgroups”, Arch. Math. (Basel), 44:1 (1985), 15–19 | MR | Zbl

[66] Berman S., Moody R., “Extensions of Chevalley groups”, Israel J. Math., 22:1 (1975), 42–51 | DOI | MR | Zbl

[67] Brown R., Humphries S. P., “Orbits under symplectic transvections, I, II”, Proc. London Math. Soc. (3), 52 (1986), 517–531 ; 532–556 | DOI | MR | Zbl

[68] Cameron P. J., Hall J. I., “Some groups generated by transvection subgroups”, J. Algebra, 140:1 (1991), 184–209 | DOI | MR | Zbl

[69] Carter R. W., “Simple groups of Lie type”, Pure Appl. Math., 28 (1972), Wiley, London | MR

[70] Cohen A. M., “Finite complex reflection groups”, Ann. Sci. École Norm. Sup. (4), 9 (1976), 379–436 | MR | Zbl

[71] Cohen A. M., “Finite quaternionic reflection groups”, J. Algebra, 64 (1980), 293–324 | DOI | MR | Zbl

[72] Cohen A. M., Cuypers H., Sterk H., “Linear groups generated by reflection tori”, Canad. J. Math., 51:6 (1999), 1149–1174 | MR | Zbl

[73] Cooperstein B. N., “Subgroups of the group $\mathrm{E}_6(q)$ which are generated by root subgroups”, J. Algebra, 46 (1977), 355–388 | DOI | MR | Zbl

[74] Cooperstein B. N., “The geometry of root subgroups in exceptional groups, I, II”, Geom. Dedicata, 8 (1979), 317–381 ; 15 (1983), 1–45 | DOI | MR | Zbl | DOI | MR | Zbl

[75] Cooperstein B. N., “Geometry of long root subgroups in groups of Lie type”, The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, CA, 1979), Proc. Sympos. Pure Math., 37, Amer. Math. Soc., Providence, RI, 1980, 243–248 | MR

[76] Cooperstein B. N., “Subgroups of exceptional groups of Lie type generated by long root elements, I, II”, J. Algebra, 70:1 (1981), 270–282 ; 283–298 | DOI | MR | Zbl | MR | Zbl

[77] Coxeter H. S. M., “Finite groups generated by unitary reflections”, Abh. Math. Sem. Univ. Hamburg, 31 (1967), 125–135 | DOI | MR | Zbl

[78] Cuypers H., “A characterization of $\mathrm{SL}_2(k)$ by its quadratic action on the natural module”, Arch. Math. (Basel), 61:5 (1993), 401–408 | MR | Zbl

[79] Cuypers H., The geometry of $k$-transvection groups, Preprint, Eindhoven Univ. Technology, 1994 | MR | Zbl

[80] Cuypers H., “Symplectic geometries, transvection groups and modules”, J. Combin. Theory Ser. A, 65 (1994), 39–59 | DOI | MR | Zbl

[81] Cuypers H., Steinbach A., “Linear transvection groups and embedded polar spaces”, Invent. Math., 137:1 (1999), 169–198 | DOI | MR | Zbl

[82] Cuypers H., Steinbach A., “Special linear groups generated by transvections and embedded projective spaces”, J. London Math. Soc. (2), 64:3 (2001), 576–594 | DOI | MR | Zbl

[83] Di Martino L., Vavilov N. A., “$(2,3)$-generation of $\mathrm{SL}(n,q)$, I, II”, Comm. Algebra, 22:4 (1994), 1321–1347 ; 24:2 (1996), 487–515 | DOI | MR | Zbl | DOI | MR | Zbl

[84] Friedland S., “Maximality of the monomial group”, Linear Multilinear Algebra, 18 (1985), 1–7 | DOI | MR | Zbl

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

[86] Harebov A. L., Vavilov N. A., “On the lattice of subgroups of Chevalley groups containing a split maximal torus”, Comm. Algebra, 24:1 (1996), 109–133 | DOI | MR | Zbl

[87] Huffman W. C., “Linear groups containing an element with an eigenspace of codimension two”, J. Algebra, 34 (1975), 260–287 | DOI | MR | Zbl

[88] Huffman W. C., Wales D. B., “Linear graphs of degree $n$ containing an element with exactly $n-2$ equal eigenvalues”, Linear Multilinear Algebra, 3:1–2 (1975–1976), 53–59 | DOI | MR | Zbl

[89] Huffman W. C., Wales D. B., “Linear groups containing an element with an eigenspace of codimension two”, Proceedings of Conference on Finite Groups (Univ. Utah, Park City, Utah, 1975), Acad. Press, New York, 1976, 425–429 | MR

[90] Huffman W. C., Wales D. B., “Linear groups containing an involution with two eigenvalues, 1”, J. Algebra, 45 (1977), 465–515 | DOI | MR | Zbl

[91] James D. G., Weisfeiler B., “On the geometry of unitary groups”, J. Algebra, 63 (1980), 514–540 | DOI | MR | Zbl

[92] Kantor W. M., “Subgroups of classical groups generated by long root elements”, Trans. Amer. Math. Soc., 248:2 (1979), 347–379 | DOI | MR | Zbl

[93] Kantor W. M., “Generation of linear groups”, The Geometric Vein: Coxeter Festschrift, Springer, New York-Berlin, 1981, 497–509 | MR

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

[95] King O. H., Subgroups of the special linear group containing the group of diagonal matrices, Preprint, Univ. Newcastle, 2005

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

[97] Li Shang Zhi, “The maximality of monomial subgroups of linear groups over division rings”, J. Algebra, 127:1 (1989), 22–39 | DOI | MR | Zbl

[98] Li Shang Zhi, “Irreducible subgroups of $\mathrm{SL}(n,K)$ generated by root subgroups”, Geom. Dedicata, 31 (1989), 41–44 | DOI | MR | Zbl

[99] Liebeck M. W., Seitz G. M., “Subgroups generated by root elements in groups of Lie type”, Ann. of Math. (2), 139 (1994), 293–361 | DOI | MR | Zbl

[100] McLaughlin J., “Some groups generated by transvections”, Arch. Math. (Basel), 18 (1967), 364–368 | MR | Zbl

[101] Mitrofanov M. Yu., Vavilov N. A., “Overgroups of the diagonal subgroup via small Bruhat cells”, Algebra Colloq. (to appear)

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

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

[104] Shephard G. C., Todd J. A., “Finite unitary reflection groups”, Canad. J. Math., 6 (1954), 274–304 | MR | Zbl

[105] Stark B., “Some subgroups of $\Omega(V)$ generated by groups of root type 1”, Illinois J. Math., 17:3 (1973), 584–607 | MR | Zbl

[106] Stark B., “Some subgroups of $\Omega(V)$ generated by groups of root type”, J. Algebra, 29:1 (1974), 33–41 | DOI | MR | Zbl

[107] Stark B., “Irreducible subgroups of orthogonal groups generated by groups of root type 1”, Pacific J. Math., 53:3 (1974), 611–625 | MR | Zbl

[108] Stark B., “Another look at Thompson's quadratic pairs”, J. Algebra, 45:2 (1977), 334–342 | DOI | MR | Zbl

[109] Steinbach A. I., Untergruppen von klassischen Gruppen, die von Transvektionen oder Siegel-Transvektionen erzeugt werden, Ph.-D. Thesis, 1995

[110] Steinbach A. I., “Subgroups of classical groups generated by transvections or Siegel transvections, I, II”, Geom. Dedicata, 68 (1997), 281–322 ; 323–357 | DOI | MR | Zbl | MR | Zbl

[111] Steinbach A. I., “Subgroups isomorphic to $\mathrm{G}_2(L)$ in orthogonal groups”, J. Algebra, 205:1 (1998), 77–90 | DOI | MR | Zbl

[112] Steinbach A. I., Groups of Lie type generated by long root elements in $\mathrm{F}_4(K)$, Habilitationsschrift, Gießen, 2000

[113] Steinbach A. I., “Subgroups of the Chevalley groups of type $\mathrm{F}_4(K)$ arising from a polar space”, Adv. Geom., 3 (2003), 73–100 | DOI | MR | Zbl

[114] Thompson J., “Quadratic pairs”, Actes du Congrès International des Mathématiciens, T. 1 (Nice, 1970), Gauthier-Villars, Paris, 1971, 375–376 | MR

[115] Timmesfeld F. G., “Groups generated by $k$-transvections”, Invent. Math., 100 (1990), 167–206 | DOI | MR | Zbl

[116] Timmesfeld F. G., “Groups generated by $k$-root subgroups”, Invent. Math., 106 (1991), 575–666 | DOI | MR | Zbl

[117] Timmesfeld F. G., “Groups generated by $k$-root subgroups – a survey, Groups”, Combinatorics and Geometry (Durham, 1990), London Math. Soc. Lecture Note Ser., 165, Cambridge Univ. Press, Cambridge, 1992, 183–204 | MR | Zbl

[118] Timmesfeld F. G., “Moufang planes and the groups $\mathrm{E}_6^K$ and $\mathrm{SL}_2(K)$, $K$ a Cayley division algebra”, Forum Math., 6:2 (1994), 209–231 | MR | Zbl

[119] Timmesfeld F. G., “Subgroups generated by root elements of groups generated by $k$-root subgroups”, Geom. Dedicata, 49 (1994), 293–321 | DOI | MR | Zbl

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

[121] Timmesfeld F. G., Abstract root subgroups and simple groups of Lie type, Monogr. in Math., 95, Birkhäuser Verlag, Basel, 2001 | MR | Zbl

[122] Vavilov N. A., “A conjugacy theorem for subgroups of $\mathrm{GL}_n$ containing the group of diagonal matrices”, Colloq. Math., 54:1 (1987), 9–14 | MR | Zbl

[123] 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

[124] Vavilov N. A., Weight elements of Chevalley groups, Preprint no. 35, Univ. Warwick, 1994, pp. 1–46

[125] Vavilov N. A., “Unipotent elements in subgroups which contain a split maximal torus”, J. Algebra, 176 (1995), 356–367 | DOI | MR | Zbl

[126] Vavilov N. A., Geometry of 1-tori in $\mathrm{GL}_n$, Preprint no. 8, Univ. Bielefeld, 1995, pp. 1–21 | Zbl

[127] Wagner A., “Groups generated by elations”, Abh. Math. Sem. Univ. Hamburg, 41 (1974), 190–205 | DOI | MR | Zbl

[128] Wagner A., “Collineation groups generated by homologies of order greater than 2”, Geom. Dedicata, 7 (1978), 387–398 | DOI | MR | Zbl

[129] Wagner A., “Determination of the finite primitive reflection groups over an arbitrary field of characteristic not 2. I–III”, Geom. Dedicata, 9 (1980), 239–253 ; 10 (1981), 191–203 ; 475–523 | DOI | MR | Zbl | DOI | MR | Zbl | Zbl

[130] Wales D. B., “Linear groups of degree n containing an involution with two eigenvalues $-1$. II”, J. Algebra, 53 (1978), 58–67 | DOI | MR | Zbl

[131] Weisfeiler B., “Abstract monomorphisms between big subgroups of some groups of type $\mathrm B_2$ in characteristic 2”, J. Algebra, 60 (1979), 204–222 | DOI | MR | Zbl

[132] Weisfeiler B., “Monomorphisms between subgroups of groups of type $\mathrm G_2$”, J. Algebra, 68 (1981), 306–334 | DOI | MR | Zbl

[133] Weisfeiler B., “Abstract isomorphisms of simple algebraic groups split by quadratic extensions”, J. Algebra, 68 (1981), 335–368 | DOI | MR | Zbl