Geodesic
On control of the prime spectrum of the finite simple groups
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 19 (2013) no. 3, pp. 29-44
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The set $\pi(G)$ of all prime divisors of the order of a finite group $G$ is often called its prime spectrum. It is proved that every finite simple nonabelian group $G$ has sections $H_1,\dots,H_m$ of some special form such that $\pi(H_1)\cup\dots\cup\pi(H_m)=\pi(G)$ and $m\le5$, in the case when $G$ is an alternating or classical simple group, in addition, $m\le2$. Moreover, in any case, it is possible to choose the sections $H_i$ so that each of them is a simple nonabelian group, a Frobenius group, or (in one case) a dihedral group. If the above equality is realized for a finite group $G$, then we say that the set $\{H_1,\dots,H_m\}$ controls the prime spectrum of $G$. We also study some parameter $c(G)$ of finite groups $G$ related to the notion of control.
Keywords: finite group, prime spectrum, maximal subgroup, section of a group.
Mots-clés : simple group 
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V. A. Belonogov. On control of the prime spectrum of the finite simple groups. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 19 (2013) no. 3, pp. 29-44. http://geodesic.mathdoc.fr/item/TIMM_2013_19_3_a3/

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

[2] Huppert B., Endliche Gruppen, v. I, Springer, Berlin, 1967, 793 pp. | MR | Zbl

[3] Wilson R. A., The finite simple groups, Springer, London, 2009, 298 pp. | MR

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

[5] Gorenstein D., Lyons R., Solomon R., The classification of the finite simple groups, Math. Surveys and Monographs, 40, no. 1, 1994, 165 pp. | MR | Zbl

[6] Belonogov V. A., “O konechnykh gruppakh, nasyschennykh $(\pi,\pi')$-razlozhimymi podgruppami”, Sib. mat. zhurn., 10:3 (1969), 494–506 | MR | Zbl

[7] Arad Z., Chillag D., “A criterium for the existence of normal $\pi$-complements in finite groups”, J. Algebra, 87:2 (1984), 472–482 | DOI | MR | Zbl

[8] Chunikhina I. K., Chunikhin S. A., “O $p$-razlozhimykh gruppakh”, Mat. sb., 15(57):2 (1944), 325–342 | MR | Zbl

[9] Vedernikov V. A., “O $\pi$-svoistvakh konechnykh grupp”, Arifmeticheskoe i podgruppovoe stroenie konechnykh grupp, Tr. Gomel. seminara In-ta matematiki AN BSSR, Minsk, 1986, 13–19 | MR | Zbl

[10] Belonogov V. A., “O podgruppakh konechnykh znakoperemennykh i klassicheskikh prostykh grupp”, Algebra i lineinaya optimizatsiya, Tez. Mezhdunar. konf., posvyasch. 100-letiyu S. N. Chernikova, Izd-vo UMTs-UPI, Ekaterinburg, 2012, 14

[11] Belonogov V. A., “Finite groups all of whose maximal subgroups are $\pi$-decomposable”, Teoriya grupp i eë prilozheniya, Tez. IX Mezhdunar. konf. po teorii grupp, posvyasch. 90-letiyu so dnya rozhdeniya prof. Z. I. Borevicha, Severo-Osetin. gos. un-t, Vladikavkaz, 2012, 126–127

[12] Huppert B., Blackburn N., Finite Groups, v. II, Springer, Berlin, 1982, 531 pp. | MR | Zbl

[13] Kleidman P., Liebeck M., The subgroup structure of the finite classical groups, London Math. Soc. Lect. Note Series, 129, Cambridge Univ. Press, Cambridge, 1990, 303 pp. | MR | Zbl

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

[15] Cooperstein B. N., “Maximal subgroups of $G_2(2^n)$”, J. Algebra, 70:1 (1981), 23–36 | DOI | MR | Zbl

[16] 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

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

[18] 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

[19] Kondratev A. S., Mazurov V. D., “Raspoznavanie znakoperemennykh grupp prostoi stepeni po poryadkam ikh elementov”, Sib. mat. zhurn., 41:2 (2000), 359–369 | MR | Zbl

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

[21] 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 (1992), 297–325 | DOI | MR | Zbl

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

[23] Shemetkov L. A., Formatsii konechnykh grupp, Nauka, M., 1978, 272 pp. | MR | Zbl