Finite simple groups in which all maximal subgroups are $\pi$-closed. II

V. A. Belonogov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 22 (2016) no. 3, pp. 12-22

Résumé

We continue the study of pairs $(G,\pi)$, where $G$ is a finite simple nonabelian group and $\pi$ a set of primes, such that $G$ has only $\pi$-closed maximal subgroups but is not $\pi$-closed itself. Using the results of the first paper from the series, we give a list of such pairs $(G,\pi)$ in the case when $G$ is different from the groups $PSL_r(q)$ and $PSU_r(q)$ with prime odd $r$ and $E_8(q)$, where $q$ is a prime power.

Détail

Keywords: finite group, $\pi$-closed group, maximal subgroup.
Mots-clés : simple group

Bibliographie

[1] Belonogov V.A., “O konechnykh gruppakh, vse maksimalnye podgruppy kotorykh $\pi$-zamknuty”, Mezhdunar. shkola-konf. po teor. grupp, posv. 70-letiyu V. V. Kabanova, sb. st., K-BGU, Nalchik, 2014, 6–9

[2] Belonogov V.A., “Konechnye prostye gruppy, vse maksimalnye podgruppy kotorykh $\pi$-zamknuty.I”, Tr. In-ta matematiki i mekhaniki UrO RAN, 21:1 (2015), 25–34 | MR

[3] Belonogov V.A., “O kontrole prostogo spektra konechnoi prostoi gruppy”, Tr. In-ta matematiki i mekhaniki UrO RAN, 19:3 (2013), 29–44 | MR

[4] Gorenstein D., Finite Groups, Harper Row, New York, 1968, 527 pp. | MR | Zbl

[5] Wilson R.A., The finite simple groups, Springer-Verlag, London, 2009, 298 pp. | MR | Zbl

[6] J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, R.A. Wilson, Atlas of finite groups, Clarendon Press, Oxford, 1985, 252 pp. | MR | Zbl

[7] Belonogov V.A., “Finite groups in which all maximal subgroups are $\pi$-closed”, Mezhdunar. konf. “Algebra i logika: Teoriya i prilozheniya”, posvyasch. 70-letiyu V.M. Levchuka, tez. dokl., Krasnoyarsk, 2016, 90

[8] Belonogov V.A., “Konechnye gruppy s tremya klassami maksimalnykh podgrupp”, Mat. sb., 131:2 (1986), 225–239 | Zbl

[9] J. Brillhart et al., Factorizations of $b^n\pm 1$, $b=2,3,5,6,7,10,11,12$ up to high powers, Contemporary Math., 22, American Math. Society, Providence, 1988, 236 p. pp. | DOI | MR

[10] Aschbacher M., Scott L., “Maximal subgroups of finite groups”, J. Algebra, 92:1 (1985), 44–80 | DOI | MR | Zbl

[11] Liebeck M.W., Praeger C.E., Saxl J., “The classification of the maximal subgroups of the finite alternating and symmetric groups”, J. Algebra, 111:2 (1987), 365–383 | DOI | MR | Zbl

[12] Guralnick R., “Subgroups of prime power index in a simple group”, J. Algebra, 81:2 (1983), 304–311 | DOI | MR | Zbl

[13] Wiman A., “Bestimmung aller Untergruppen einer doppelt unendlichen Reihe von einfachen Gruppen”, Stockh. Acad. Bihang., 25:2 (1899), 1–47 | MR | Zbl

[14] Moor E.H., “The subgroups of the generalized finite modular group”, Dicennial Publications of the University of Chicago, 9, The University of Chicago Press, Chicago, 1903, 141–190

[15] King O.H., “The subgroup structure of finite classical groups in terms of geometric configuratios”, Surveys in combinatorics, London Math. Soc. Lecture Note Ser., 327, Cambridge Univ. Press, Cambridge, 2005, 29–56 | MR | Zbl

[16] Bray J.N., Holt D.F., Roney-Dougal C.M., The maximal subgroups of the low-dimensional finite classical groups, London Math. Soc. Lect. Note Ser., 407, Cambridge Univ. Press, Cambridge, 2013, 438 pp. | MR | Zbl

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

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

[19] Kleidman P.B., “The maximal subgroups of the finite Chevalley groups $G_2(q)$ with $q$ odd, the Ree groups ${}^2G_2(q)$, and their automorphism groups”, J. Algebra, 117:1 (1988), 30–71 | DOI | MR | Zbl

[20] Kleidman P.B., “The maximal subgroups of the Steinberg triality groups ${}^3D_4(q)$ and their automorphism groups”, J. Algebra, 115:1 (1988), 182–199 | DOI | MR | Zbl

[21] Malle G., “The maximal subgroups of ${}^2F_4(q)$”, J. Algebra, 139:1 (1991), 52–69 | DOI | MR | Zbl