Finite Groups Whose Maximal Subgroups Are Solvable or Have Prime Power Indices

Guo Wen Bin ; A. S. Kondrat'ev ; N. V. Maslova ; L. Miao

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 2, pp. 125-131

Résumé

It is well known that all maximal subgroups of a finite solvable group are solvable and have prime power indices. However, the converse statement does not hold. Finite nonsolvable groups in which all local subgroups are solvable were studied by J. Thompson (1968). R. Guralnick (1983) described all the pairs $(G,H)$ such that $G$ is a finite nonabelian simple group and $H$ is a subgroup of prime power index in $G$. Several authors studied finite groups in which every subgroup of non-prime-power index (not necessarily maximal) is a group close to nilpotent. Weakening the conditions, E. N. Bazhanova (Demina) and N. V. Maslova (2014) considered the class $\mathfrak{J}_{\rm pr}$ of finite groups in which all nonsolvable maximal subgroups have prime power indices and, in particular, described possibilities for nonabelian composition factors of a nonsolvable group from the class $\mathfrak{J}_{\rm pr}$. In the present note, the authors continue the study of the normal structure of a nonsolvable group from $\mathfrak{J}_{\rm pr}$. It is proved that a group from $\mathfrak{J}_{\rm pr}$ contains at most one nonabelian chief factor and, for each positive integer $n$, there exists a group from $\mathfrak{J}_{\rm pr}$ such that the number of its nonabelian composition factors is at least $n$. Moreover, all almost simple groups from $\mathfrak{J}_{\rm pr}$ are determined.

Détail

Keywords: finite group, maximal subgroup, prime power index, nonsolvable subgroup.

Bibliographie

[1] Demina E.N., Maslova N.V., “Neabelevy kompozitsionnye faktory konechnoi gruppy s arifmeticheskimi ogranicheniyami na nerazreshimye maksimalnye podgruppy”, Tr. In-ta matematiki i mekhaniki UrO RAN, 20:2 (2014), 122–134

[2] Go V., Maslova N.V., Revin D.O., “O pronormalnosti podgrupp nechetnykh indeksov v nekotorykh rasshireniyakh konechnykh grupp”, Sib. mat. zhurn., 59:4 (2018), 773–790 | MR | Zbl

[3] Kondratev A.S., “O komponentakh grafa prostykh chisel konechnykh prostykh grupp”, Mat. sb., 180:6 (1989), 787–797 | Zbl

[4] Maslova N.V., “Maksimalnye podgruppy nechetnogo indeksa v konechnykh gruppakh s prostym lineinym, unitarnym ili simplekticheskim tsokolem”, Algebra i logika, 50:2 (2011), 189–208 | MR | Zbl

[5] Baryshovets P.P., “Finite nonsolvable groups in which subgroups of nonprimary index are nilpotent or are Shmidt groups”, Ukrain. Math. J., 33:1 (1981), 37–39 | DOI | MR | Zbl

[6] 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 University Press, Cambridge, 2013, 438 pp. | DOI | MR | Zbl

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

[8] Giudici M., Maximal subgroups of almost simple groups with socle $PSL(2, q)$, [e-resource], 2007, 11 pp., arXiv: math/0703685

[9] Gorenstein D., Finite groups, Chelsea, N Y, 1968, 520 pp. | MR

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

[11] Huppert B., “Singer-Zyklen in klassischen Gruppen”, Math. Z., 117:7 (1970), 141–150 | DOI | MR | Zbl

[12] Kleidman P., Liebeck M., The Subgroup structure of the finite classical groups, Cambridge Univ. Press., Cambridge, 1990, 304 pp. | MR | Zbl

[13] Thompson J.G., “Nonsolvable finite groups all of whose local subgroups are solvable”, Bull. Amer. Math. Soc., 74:3 (1968), 383–437 | DOI | MR | Zbl