2-Signalizers of Finite Simple Groups

A. S. Kondrat'ev ; V. D. Mazurov

Algebra i logika, Tome 42 (2003) no. 5, pp. 594-623.

Résumé

Maximal 2-signalizers and centralizers of Sylow 2-subgroups in all finite simple groups are described. Also normalizers are computed for Sylow 2-subgroups in the finite simple groups of exceptional Lie type over a field of odd characteristic.

Détail

Keywords: finite simple group, finite simple group of exceptional Lie type over a field of odd characteristic, Sylow 2-subgroup, maximal 2-signalizer, centralizer, normalizer.

Bibliographie

[1] J. G. Thompson, “2-signalizers of finite groups”, Pac. J. Math., 14:1 (1964), 363–364 | MR | Zbl

[2] V. D. Mazurov, “O 2-signalizatorakh konechnykh grupp”, Algebra i logika, 7:3 (1968), 60–62 | MR | Zbl

[3] D. Gorenstein, Finite simple groups. An introduction to their classification, Plenum Press, New York, 1982 | MR | Zbl

[4] Z. Janko, J. G. Thompson, “On a class of finite simple groups of Ree”, J. Algebra, 4:2 (1966), 274–292 | DOI | MR | Zbl

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

[6] M. W. Liebeck, J. Saxl, “The primitive permutation groups of odd degree”, J. Lond. Math. Soc., II. Ser., 31:2 (1985), 250–264 | DOI | MR | Zbl

[7] W. M. Kantor, “Primitive permutation groups of odd degree, and an application to finite projective planes”, J. Algebra, 106:1 (1987), 15–45 | DOI | MR | Zbl

[8] M. Aschbacher, “On finite groups of Lie type and odd characteristic”, J. Algebra, 66:2 (1980), 400–424 | DOI | MR | Zbl

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

[10] M. Aschbacher, Finite group theory, Cambridge University Press, Cambridge, 1986 | MR | Zbl

[11] R. W. Carter, Simple groups of Lie type, Wiley, London, 1972 | MR | Zbl

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

[13] Seminar po algebraicheskim gruppam, sb. statei, ed. A. A. Kirillova, Mir, M., 1973 | MR

[14] M. Aschbacher, “A characterization of Chevalley groups over fields of odd order”, Ann. Math. (2), 106:2–3 (1977), 353–468 ; Correction, Ann. Math. (2), 111:3 (1980), 411–414 | DOI | MR | Zbl | DOI | MR | Zbl

[15] M. E. Harris, “Finite groups containing an intrinsic 2-component of Chevalley type over a field of odd order”, Trans. Am. Math. Soc., 272:1 (1982), 1–65 | DOI | MR | Zbl

[16] M. W. Liebeck, J. Saxl, G. M. Seitz, “Subgroups of maximal rank in finite exceptional groups of Lie type”, Proc. Lond. Math. Soc., III. Ser., 65:2 (1985), 297–325 | DOI | MR

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

[18] G. M. Seitz, “Flag-transitive subgroups of Chevalley groups”, Ann. Math., 97:1 (1973), 27–56 | DOI | MR | Zbl

[19] A. Borel, J. Tits, “ElNments unipotents et sousgroupes paraboliques de groupes réductifs. I”, Invent. Math., 12:2 (1971), 95–104 | DOI | MR | Zbl

[20] D. Gorenstein, R. Lyons, The local structure of finite groups of characteristic 2 type, Mem. Am. Math. Soc., 42, no. 276, Am. Math. Soc., Providence, RI, 1983 | MR

[21] M. Aschbacher, Overgroups of Sylow subgroups in sporadic groups, Mem. Am. Math. Soc., 60, no. 343, Am. Math. Soc., Providence, RI, 1986 | MR

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

[23] L. E. Dickson, Linear groups with an exposition of the Galois field theory, Dover, New York, 1958 | MR | Zbl

[24] M. Aschbacher, “On the maximal subgroups of the finite classical groups”, Invent. Math., 76:3 (1984), 469–514 | DOI | MR | Zbl

[25] P. B. Kleidman, “The maximal subgroups of the finite 8-dimensional orthogonal groups $P\Omega_8^+(q)$ and of their automorphism groups”, J. Algebra, 66:1 (1987), 173–242 | DOI | MR

[26] D. Gorenstein, K. Harada, “Finite simple groups of low rank and the families $G_2(q)$, $D_4^2(q)$, $q$ odd”, Bull. Am. Math. Soc., 77:6 (1971), 829–862 | DOI | MR | Zbl