Finite groups with four conjugacy classes of maximal subgroups. III

V. A. Belonogov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 27 (2021) no. 1, pp. 5-18

Résumé

We continue the study of finite groups with exactly four conjugacy classes of maximal subgroups. Groups with this property are called $4M$-groups. In the first part of this series of papers, we described simple $4M$-groups and nonsimple nonsolvable $4M$-groups without normal subgroups of prime index. In the second part, we started the investigation of finite nonsolvable $4M$-groups with a normal maximal subgroup using G. Pazderski's results on the structure of finite groups with exactly two conjugacy classes of maximal subgroups and the author's results on the structure of finite groups with exactly three conjugacy classes of maximal subgroups. The results of parts I and II are recalled in the introduction in Theorems 1–3. In the present third part, a complete description of finite nonsimple almost simple $4M$-groups is given (see Theorem 4). The proofs of the results are based on the works of many authors who studied the structure of maximal subgroups of finite simple and almost simple groups from various classes.

Détail

Keywords: finite group, almost simple group, maximal subgroup
Mots-clés : $4M$-group.

Bibliographie

[1] Belonogov V.A., “Konechnye gruppy s chetyrmya klassami maksimalnykh podgrupp. I”, Tr. In-ta matematiki i mekhaniki UrO RAN, 23:4 (2017), 52–62 | DOI | MR

[2] Belonogov V.A., “Konechnye gruppy s chetyrmya klassami maksimalnykh podgrupp. II”, Sib. elektron. mat. izv., 15 (2018), 86–91 | DOI | Zbl

[3] Belonogov V.A., “Konechnye gruppy s chetyrmya klassami maksimalnykh podgrupp”, Teoriya grupp i ee prilozheniya, materialy XII shk.-konf. po teorii grupp, posvyaschen. 65-letiyu A. A. Makhneva, Kubanskii gos. un-t, Krasnodar, 2018, 7–13

[4] Pazderski G., “Uber maximal Untergruppen endlicher Gruppen”, Math. Nachr., 26:6 (1964), 307–319 | MR | Zbl

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

[6] Belonogov V.A., “On finite almost simple groups with exactly four conjugate classes of maximal subgroups”, Tez. Mezhdunar. konf. “Maltsevskie chteniya” (Malcev Meeting), 2019, 142

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

[8] Gorenstein D., Finite groups, Harper Row, N Y, 1968, 642 pp. | MR | Zbl

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

[10] Wilson R.A., The finite simple groups, Springer, London, 2009, 313 pp. | 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] Bray J.N., Holt D.F., Roney-Dougal C.M., The maximal subgroups of the low-dimensional finite classical groups, London Math. Soc. Lecture Note, 407, Cambridge Univ. Press, Cambridge, 2013, 438 pp. | MR | Zbl

[13] Huppert B., Endliche Gruppen, v. 1, Springer, Berlin, 1967, 793 pp. | DOI | MR | Zbl

[14] Dickson L.E., Linear groups with an exposition of the Galois field theory, Dover reprint 1958, Teubner, Leipzig, 1901 | DOI | MR | Zbl

[15] King O., “The subgroup structure of finite classical groups in terms of geometric configurations”, Surveys in Combinatorics, British Combinatorial Conf. - BC, 2005, London Math. Soc. Lecture Note Ser., 327, ed. B.S. Webb, Cambridge Univ. Press, Cambridge, 2005, 29–56 | DOI | MR | Zbl

[16] Giudici M., The maximal subgroups of almost simple groups with socle $PSL(2,q)$, [e-resource], 2007, 11 pp., arXiv: math.0703685v1

[17] Kleidman P., Liebeck M., The subgroup structure of the finite classical groups, London Math. Soc. Lecture Note Ser., 129, Cambridge Univ. Press, Cambridge, 1990, 304 pp. | MR | Zbl

[18] Carter R.W., Simple groups of Lie type, John Willey and Sons, London, 1972, 331 pp. | MR | Zbl

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

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