On primitive permutation groups with the stabilizer of two points normal in the stabilizer of one of them: The case when the socle is a power of a group $E_8(q)$

A. V. Konygin

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 25 (2019) no. 4, pp. 88-98

Résumé

Assume that $G$ is a primitive permutation group on a finite set $X$, $x\in X\setminus\{x\}$, and $G_{x, y}\trianglelefteq G_x$. P. Cameron raised the question about the validity of the equality $G_{x, y} = 1$ in this case. The author proved earlier that, if the socle of $G$ is not a power of a group isomorphic to $E_8(q)$ for a prime power $q$, then $G_{x, y}=1$. In the present paper, we consider the case where the socle of $G$ is a power of a group isomorphic to $E_8(q)$. Together with the previous result, we establish the following two statements. 1. Let $G$ be an almost simple primitive permutation group on a finite set $X$. Assume that, if the socle of $G$ is isomorphic to $E_8(q)$, then $G_x$ for $x \in X$ is not the Borovik subgroup of $G$. Then the answer to Cameron's question for such primitive permutation groups is affirmative. 2. Let $G$ be a primitive permutation group on a finite set $X$ with the property $G\leq H\mathrm{ wr } S_m$. Assume that, if the socle of $H$ is isomorphic to $E_8(q)$, then the stabilizer of a point in the group $H$ is not the Borovik subgroup of $H$. Then the answer to Cameron's question for such primitive permutation groups is also affirmative.

Détail

Mots-clés : primitive permutation group
Keywords: regular suborbit.

Bibliographie

[1] Benson D., Carlson J., “Nilpotent elements in the Green ring”, J. Algebra, 104 (1986), 329–350 | DOI | MR | Zbl

[2] Borovik A. V., “A maximal subgroup in the simple finite group $E_8(q)$”, Contemporary Mathematics, 131:1 (1992), 67–79 | DOI | MR | Zbl

[3] Cameron P.J., “Suborbits in transitive permutation groups”, Combinatorics. Part 3: Combinatorial Group Theory, Proc. NATO Advanced Study Inst. (Breukelen, 1974), Math. Centre Tracts, 57, Math. Centrum, Amsterdam, 1974, 98–129 | MR

[4] Cohen A.M., Liebeck M.W., Saxl J., Seitz G.M., “The local maximal subgroups of exceptional groups of Lie type, finite and algebraic”, Proc. London Mat. Soc. (3), 64 (1992), 21–48 | DOI | MR

[5] Conway J.H. et. al., Atlas of finite groups, Clarendon Press, Oxford, 1985, 252 pp. | MR | Zbl

[6] Craven D.A., “On tensor products of simple modules for simple groups”, Algebras and Representation Theory, 16:2 (2013), 377–404 | DOI | MR | Zbl

[7] Doty S., Henke A., “Decomposition of tensor products of modular irreducibles for SL${}_2$”, Q. J. Math. Algebra, 56:2 (2005), 189–207 | DOI | MR | Zbl

[8] Feit W., The representation theory of finite groups, North-Holland Mathematical Library, Elsevier, North-Holland, 1982, 501 pp. | MR

[9] Fomin A.N., “Svoistva podorbit konechnykh primitivnykh grupp podstanovok”, Teoretiko-gruppovye issledovaniya, sb. nauch. tr., UrO AN SSSR, Sverdlovsk, 1990, 87–94

[10] Humphreys J.E., Modular representations of finite groups of Lie type, Cambridge University Press, Cambridge, 2011, 206 pp. | DOI | MR

[11] Konygin A.V., “O primitivnykh gruppakh podstanovok so stabilizatorom dvukh tochek, normalnym v stabilizatore odnoi iz nikh”, Sib. elektron. mat. izv., 5 (2008), 387–406 | MR | Zbl

[12] Konygin A.V., “K voprosu P. Kamerona o primitivnykh gruppakh podstanovok so stabilizatorom dvukh tochek, normalnym v stabilizatore odnoi iz nikh”, Tr. In-ta matematiki i mekhaniki UrO RAN, 19:3 (2013), 187–198

[13] Knapp W., “Primitive Permutationsgruppen mit einem Subkonstituenten, dessen Stabilisatorgruppe Fittingfrei ist”, Arch. Math., 25 (1974), 472–475 | DOI | MR | Zbl

[14] Knapp W., “Some problems of Wielandt revisited”, J. Algebra, 302:1 (2006), 167–185 | DOI | MR | Zbl

[15] Liebeck M.W., Praeger Ch.E., Saxl J., “On the O'Nan-Scott theorem for finite primitive permutation groups”, J. Austral. Math. Soc. Ser. A, 44:3 (1988), 389–396 | DOI | MR | Zbl

[16] Liebeck M.W., Saxl J., Seitz G.M., “Subgroups of maximal rank in finite exceptional groups of Lie type”, Proc. London Math. Soc. (3), 65:2 (1992), 297–325 | DOI | MR | Zbl

[17] Liebeck M.W., Seitz G.M., “Maximal subgroups of exceptional groups of Lie type, finite and algebraic”, Geom. Dedicata, 35:1–3 (1990), 353–387 | DOI | MR | Zbl

[18] Liebeck M.W., Seitz G.M., The maximal subgroups of positive dimension in exceptional algebraic groups, Mem. Amer. Math. Soc., 169, no. 802, AMS, Providence, 2004, 227 pp. | MR

[19] Lubeck F., “Small degree representations of finite Chevalley groups in defining characteristic”, LMS J. Comput. Math., 2001, no. 4, 135–169 | DOI | MR | Zbl

[20] Malle G., Testerman D., Linear algebraic groups and finite groups of Lie type, Cambridge Univ. Press, Cambridge, 2011, 309 pp. | DOI | MR | Zbl

[21] Reitz H.L., “On primitive groups of odd order”, Amer. J. Math., 26 (1904), 1–30 | DOI | MR

[22] Seitz G.M., Maximal subgroups of exceptional algebraic groups, Mem. Amer. Math. Soc., 90, no. 441, AMS, Providence, 1991, 197 pp. | MR

[23] E.I. Khukhro, V.D. Mazurov, Unsolved problems in group theory. The Kourovka Notebook, [e-resource]June 2018, 248 pp., arXiv: 1401.0300v13 | MR

[24] Weiss M.J., “On simply transitive groups”, Bull. Amer. Math. Soc., 40 (1934), 401–405 | DOI | MR | Zbl

[25] Wielandt H., Finite permutation groups, Acad. Press, N Y, 1964, 114 pp. | MR | Zbl