Carter subgroups of finite groups

E. P. Vdovin

Matematičeskie trudy, Tome 11 (2008) no. 2, pp. 20-106.

Résumé

It is proven that the Carter subgroups of a finite group are conjugate. A complete classification of the Carter subgroups in finite almost simple groups is also obtained.

Bibliographie

[1] Borel A. i dr., Seminar po algebraicheskim gruppam, Mir, M., 1973 | MR | Zbl

[2] Vavilov N. A., “Nilpotentnye podgruppy v polnoi lineinoi gruppe nad konechnym polem, sovpadayuschie so svoim normalizatorom”, Zap. nauch. seminarov LOMI, 86, Nauka, L., 1979, 34–39 | MR | Zbl

[3] Vdovin E. P., “O probleme sopryazhennosti dlya karterovykh podgrupp”, Sib. mat. zhurn., 47:4 (2006), 725–730 | MR | Zbl

[4] Vdovin E. P., “Karterovy podgruppy v konechnykh pochti prostykh gruppakh”, Algebra i logika, 46:2 (2007), 157–216 | MR

[5] Vdovin E. P., “Karterovy podgruppy konechnykh grupp”, Dokl. AN SSSR, 415:3 (2007), 300–303

[6] Vdovin E. P., “O suschestvovanii karterovykh podgrupp”, Trudy IMM UrO RAN, 13:1 (2007), 79–88 | MR

[7] Vdovin E. P., Galt A. A., “Normalizatory podsistemnykh podgrupp v konechnykh gruppakh lieva tipa”, Algebra i logika, 47:1 (2008), 3–30 | MR

[8] Dynkin E. B., “Poluprostye podalgebry poluprostykh algebr Li”, Mat. sb., 30(72):2 (1952), 349–462 | MR | Zbl

[9] Kargapolov M. I., Merzlyakov Yu. I., Osnovy teorii grupp, Nauka, M., 1996 | MR | Zbl

[10] Kondratev A. S., “Normalizatory silovskikh 2-podgrupp konechnykh prostykh grupp”, Mat. zametki, 78:3 (2005), 368–376 | MR | Zbl

[11] Kondratev A. S., Mazurov V. D., “2-signalizatory konechnykh prostykh grupp”, Algebra i logika, 42:5 (2003), 594–623 | MR | Zbl

[12] Levchuk V. M., Nuzhin Ya. N., “O stroenii grupp Ri”, Algebra i logika, 24:1 (1985), 26–41 | MR | Zbl

[13] Khamfri Dzh., Lineinye algebraicheskie gruppy, Nauka, M., 1980 | MR

[14] Borel A., de Siebental J., “Les-sous-groupes fermés de rang maximum des groupes de Lie clos”, Comment. Math. Helv., 23 (1949), 200–221 | DOI | MR | Zbl

[15] Carter R. W., “Nilpotent self-normalizing subgroups of soluble groups”, Math. Z., 75 (1961), 136–139 | DOI | MR | Zbl

[16] Carter R. W., “Conjugacy classes in the Weyl group”, Compositio Math., 25:1 (1972), 1–59 | MR | Zbl

[17] Carter R. W., Simple Groups of Lie Type, John Wiley and Sons, New York, 1972 | MR | Zbl

[18] Carter R. W., “Centralizers of semisimple elements in finite groups of Lie type”, Proc. London Math. Soc. (3), 37:3 (1978), 491–507 | DOI | MR | Zbl

[19] Carter R. W., “Centralizers of semisimple elements in the finite classical groups”, Proc. London Math. Soc. (3), 42:1 (1981), 1–41 | DOI | MR | Zbl

[20] Carter R. W., Finite Groups of Lie Type, Conjugacy Classes, and Complex Characters, John Wiley and Sons, New York, 1985 | MR | Zbl

[21] Carter R. W., Fong P., “The Sylow 2-subgroups of the finite classical groups”, J. Algebra, 1 (1964), 139–151 | DOI | MR | Zbl

[22] Conway J. H., Curtis R. T., Norton S. P., Parker R. A., Wilson R. A., Atlas of Finite Groups, Clarendon Press, Oxford, 1985 | MR | Zbl

[23] Dalla V. F., Lucchini A., Tamburini M. C., “On the conjugacy problem for Carter subgroups”, Comm. Algebra, 26:2 (1998), 395–401 | DOI | MR

[24] D'aniello A., “Sull' esistenza di sottogruppi nilpotenti autonormalizzanti in alcuni gruppi semplici, II”, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 74 (1983), 1–6 | MR

[25] Deriziotis D., Conjugacy Classes and Centralizers of Semisimple Elements in Finite Groups of Lie Type, Vorlesungen aus dem Fachbereich Mathematic der Universität Essen, 11, Universität Essen, Fachbereich Mathematic, Essen, 1984 | MR | Zbl

[26] Di Martino L., Tamburini M. C., “I sottogruppi nilpotenti autonormalizzanti di $S_n$ e di $A_n$”, Istit. Lombardo Accad. Sci. Lett. Rend. A, 110:2 (1976), 235–241 | MR

[27] Di Martino L., Tamburini M. C., “I sottogruppi nilpotenti autonormalizzanti di $\operatorname{GL}_n(q)$”, Istit. Lombardo Accad. Sci. Lett. Rend. A, 110 (1976), 589–596 | MR

[28] Di Martino L., Tamburini M. C., “Carter subgroups of projective linear groups”, Boll. Un. Mat. Ital. B (7), 1:3 (1987), 905–915 | MR | Zbl

[29] Di Martino L., Tamburini M. C., Zalesski A. E., “Carter subgroups in classical groups”, J. London Math. Soc. (2), 55:2 (1997), 264–276 | DOI | MR | Zbl

[30] Enomoto H., “The conjugacy classes of Chevalley groups of type ($G_2$) over finite fields of characteristic 2 or 3”, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 16:3 (1970), 497–512 | MR | Zbl

[31] Feit W., Thompson J., “Solvability of groups of odd order”, Pacific J. Math., 13:3 (1963), 775–1029 | MR | Zbl

[32] Feit W., Zuckerman G., “Reality properties of conjugacy classes in spin groups and symplectic groups”, Algebraists' Homage: Papers in Ring Theory and Related Topics (New Haven, 1981), Contemp. Math., 13, Amer. Math. Soc., Providence, RI, 1982, 239–253 | MR

[33] Gorenstein D., Lyons R., The Local Structure of Finite Groups of Characteristic 2 Type, Mem. Amer. Math. Soc., 42, no. 276, Amer. Math. Soc., Providence, RI, 1983 | MR

[34] Gorenstein D., Lyons R., Solomon R., The Classification of the Finite Simple Groups. Number 3. Part I. Chapter A. Almost Simple $K$-Groups, Mathematical Surveys and Monographs, 40.3, Amer. Math. Soc., Providence, RI, 1998 | MR | Zbl

[35] Hartley B., Shute G., “Monomorphisms and direct limits of finite groups of Lie type”, Quart. J. Math. Oxford Ser. (2), 35:137 (1984), 49–71 | DOI | MR | Zbl

[36] Humphreys J. E., Conjugacy Classes in Semisimple Algebraic Groups, Mathematical Surveys and Monographs, 43, Amer. Math. Soc., Providence, RI, 1995 | MR | Zbl

[37] Menegazzo F., Tamburini M. C., A property of the normalizers of Sylow $p$-subgroups in simple groups, Quad2002n45, Sem. Mat. Brescia, 2002

[38] Mizuno K., “The conjugate classes of Chevalley groups of type $E_6$”, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24:3 (1977), 525–563 | MR | Zbl

[39] Previtali A., Tamburini M. C., Vdovin E. P., “The Carter subgroups of some classical groups”, Bull. London Math. Soc., 36:2 (2004), 145–155 | DOI | MR | Zbl

[40] Robinson D. J. S., A Course in the Theory of Groups, Graduate Texts in Mathematics, 80, Springer-Verlag, New York, 1996 | MR

[41] Seitz G. M., “Unipotent elements, tilting modules, and saturation”, Invent. Math., 141:3 (2000), 467–502 | DOI | MR | Zbl

[42] Shoji T., “The conjugacy classes of Chevalley groups of type ($F_4$) over finite fields of characteristic $p\ne2$”, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 21:1 (1974), 1–17 | MR | Zbl

[43] Steinberg R., “Automorphisms of finite linear groups”, Canad. J. Math., 12:4 (1960), 606–615 | MR | Zbl

[44] Steinberg R., Endomorphisms of Algebraic Groups, Mem. Amer. Math. Soc., 80, Amer. Math. Soc., Providence, RI, 1968 | MR

[45] Tamburini M. C., Vdovin E. P., “Carter subgroups of finite groups”, J. Algebra, 255:1 (2002), 148–163 | DOI | MR | Zbl

[46] Tiep P. H., Zalesski A. E., “Real conjugacy classes in algebraic groups and finite groups of Lie type”, J. Group Theory, 8:3 (2005), 291–315 | DOI | MR | Zbl

[47] Ward H. N., “On Ree's series of simple groups”, Trans. Amer. Math. Soc., 121:1 (1966), 62–80 | DOI | MR

[48] Wonenburger M. J., “Transformations which are products of involutions”, J. Math. Mech., 16:1 (1966), 32–38 | MR