Maximal Orders of Abelian Subgroups in Finite Chevalley Groups

E. P. Vdovin

Matematičeskie zametki, Tome 69 (2001) no. 4, pp. 524-549.

Résumé

In the present paper, for any finite group $G$ of Lie type (except for ${}^2F_4(q)$), the order $a(G)$ of its large Abelian subgroup is either found or estimated from above and from below (the latter is done for the groups $F_4(q)$, $E_6(q)$, $E_7(q)$, $E_8(q)$ and ${}^2E_6(q^2)$). In the groups for which the number $a(G)$ has been found exactly, any large Abelian subgroup coincides with a large unipotent or a large semisimple Abelian subgroup. For the groups $F_4(q)$, $E_6(q)$, $E_7(q)$, $E_8(q)$ and ${}^2E_6(q^2)$, it is shown that if an Abelian subgroup contains a noncentral semisimple element, then its order is less than the order of an Abelian unipotent group. Hence in these groups the large Abelian subgroups are unipotent, and in order to find the value of $a(G)$ for them, it is necessary to find the orders of the large unipotent Abelian subgroups. Thus it is proved that in a finite group of Lie type (except for ${}^2F_4(q)$) any large Abelian subgroup is either a large unipotent or a large semisimple Abelian subgroup.

Bibliographie

[1] 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

[2] 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

[3] Deriziotis D. I., Conjugacy classes and centralizers of semisimple elements in finite groups of Lie type, Vorlesungen aus dem Fachbereich Mathematik, 11, Universität Essen, 1984 | MR | Zbl

[4] Deriziotis D. I., “The centralizers of semisimple elements of the Chevalley groups $E_7$ and $E_8$”, Tokyo J. Math., 6:1 (1983), 191–216 | MR | Zbl

[5] Barry M. J. J., “Large Abelian subgroups of Chevalley groups”, J. Austral. Math. Soc. Ser. A, 27:1 (1979), 59–87 | DOI | MR | Zbl

[6] Barry M. J. J., Wong W. J., “Abelian $2$-subgroups of finite symplectic groups in characteristic $2$”, J. Austral. Math. Soc. Ser. A, 33:3 (1982), 345–350 | DOI | MR | Zbl

[7] Wong W. J., “Abelian unipotent subgroups of finite orthogonal groups”, J. Austral. Math. Soc. Ser. A, 32:2 (1982), 223–245 | DOI | MR | Zbl

[8] Wong W. J., “Abelian unipotent subgroups of finite unitary and symplectic groups”, J. Austral. Math. Soc. Ser. A, 33:3 (1982), 331–344 | DOI | MR | Zbl

[9] Gorenstein D., Konechnye prostye gruppy. Vvedenie v ikh klassifikatsiyu, Mir, M., 1985 | Zbl

[10] Carter R. W., Simple Groups of Lie Type, Wiley, London, 1972 | Zbl

[11] Khamfri Dzh., Lineinye algebraicheskie gruppy, Nauka, M., 1980

[12] Humphreys J. E., Conjugacy Classes in Semisimple Algebraic Groups, Math. Survey and Monographs, 43, Amer. Math. Soc., Providence (R.I.), 1995 | MR | Zbl

[13] Gorenstein D., Lyons R., The Local Structure of Finite Groups of Characteristic $2$ Type, Memoirs Amer. Math. Soc., 276, Amer. Math. Soc., Providence (R.I.), 1983

[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] Steinberg R., Endomorphisms of Linear Algebraic Groups, Memoirs Amer. Math. Soc., 80, Amer. Math. Soc., Providence (R.I.), 1968 | MR

[16] Borel A., Linear Algebraic Groups, Benjamin, New York, 1969 | Zbl

[17] Vdovin E. P., “Poryadki abelevykh podgrupp v konechnykh prostykh gruppakh”, Algebra i logika, 38:2 (1999), 131–160 | MR | Zbl

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

[19] Suzuki M., “On a class of doubly transitive groups”, Ann. Math., 75:1 (1962), 105–145 | DOI | MR | Zbl

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

[21] Seminar po algebraicheskim gruppam, Sb. statei, Mir, M., 1973