On nonabelian composition factors of a finite group that is prime spectrum minimal

N. V. Maslova   ; D. O. Revin

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 19 (2013) no. 4, pp. 155-166

Résumé

Suppose that $L$ is a finite group, $\pi(L)$ is the set of prime divisors of the order $|L|$, and $\mathfrak{Y}$ is the class of finite groups $G$ such that $\pi(G) \not = \pi(H)$ for any proper subgroup $H$ of $G$. Groups from the class $\mathfrak{Y}$ will be called prime spectrum minimal. Many but not all finite simple groups are prime spectrum minimal. For finite simple groups not from the class $\mathfrak{Y}$, the question whether they are isomorphic to nonabelian composition factors of groups from the class $\mathfrak{Y}$ is interesting. We describe some finite simple groups that are not isomorphic to nonabelian composition factors of groups from the class $\mathfrak{Y}$ and construct an example of a finite group from $\mathfrak{Y}$ that has as its composition factor a finite simple sporadic McLaughlin group $McL$ not from the class $\mathfrak{Y}$.

Détail

Keywords: finite group, prime spectrum, minimal group, maximal subgroup, composition factor.

Bibliographie

[1] Kondratev A. S., Gruppy i algebry Li, Izd-vo IMM UrO RAN, Ekaterinburg, 2009, 310 pp.

[2] Maslova N. V., “Neabelevy kompozitsionnye faktory konechnoi gruppy, vse maksimalnye podgruppy kotoroi khollovy”, Sib. mat. zhurn., 53:5 (2012), 1065–1076 | MR | Zbl

[3] Maslova N. V., Revin D. O., “Konechnye gruppy, v kotorykh vse maksimalnye podgruppy khollovy”, Mat. tr., 15:2 (2012), 105–126 | MR

[4] Maslova N. V., Revin D. O., “Porozhdaemost konechnoi gruppy s khollovymi maksimalnymi podgruppami paroi sopryazhennykh elementov”, Tr. In-ta matematiki i mekhaniki UrO RAN, 19, no. 3, 2013, 199–206

[5] Monakhov V. S., “Konechnye $\pi$-razreshimye gruppy s khollovymi maksimalnymi podgruppami”, Mat. zametki, 84:3 (2008), 390–394 | DOI | MR | Zbl

[6] Kourovskaya tetrad. Nereshennye voprosy teorii grupp, Izd. 17-e, dop., In-t matematiki SO RAN, Novosibirsk, 2010, 219 pp. http://math.nsc.ru/alglog/17kt.pdf

[7] Tikhonenko T. V., Tyutyanov V. N., “Konechnye gruppy s maksimalnymi khollovymi podgruppami”, Izv. Gomel. gos. un-ta im. F. Skoriny, 50:5 (2008), 198–206

[8] Aschbacher M., Finite group theory, Cambridge Univ. Press, Cambridge, 1986, 274 pp. | MR | Zbl

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

[10] Bray J. N., Holt D. F., Roney-Dougal C. M., The maximal subgroups of the low-dimensional finite classical groups, Cambridge Univ. Press, Cambridge, 2013, 438 pp. | MR | Zbl

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

[12] Dixon J. D., Mortimer B., Permutation groups, Springer-Verlag, N. Y., 1996, 353 pp. | MR | Zbl

[13] Gorenstein D., Finite groups, Chelsea Publishing Company, N. Y., 1968, 519 pp. | MR

[14] Kleidman P., Liebeck M., The subgroup structure of the finite classical groups, Cambridge University Press, Cambridge, 1990, 303 pp. | MR | Zbl

[15] Liebeck M. W., Praeger C. E., Saxl J., “Transitive subgroups of primitive permutation groups”, J. Algebra, 234:2 (2000), 291–361 | DOI | MR | Zbl

[16] Robinson D., A course in the theory of groups, Springer-Verlag, N. Y., 1996, 499 pp. | MR

[17] Zavernitsine A. V., “Subextensions for a permutation $PSL_2(q)$-module”, Siber. Electron. Math. Rep., 10 (2013), 551–557