Examples of Nonpronormal Relatively Maximal Subgroups of Finite Simple Groups

B. Li ; D. O. Revin

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 29 (2023) no. 4, pp. 140-145

Résumé

Using R. Wilson's recent results, we prove the existence of triples $(\mathfrak{X},G,H)$ such that $\mathfrak{X}$ is a complete (i.e., closed under taking subgroups, homomorphic images, and extensions) class of finite groups, $G$ is a finite simple group, and $H$ is its $\mathfrak{X}$-maximal subgroup nonpronormal in $G$. This disproves a conjecture stated earlier by the second author and W. Guo.

Détail

Keywords: complete class of groups, relatively maximal subgroup, pronormal subgroup, finite simple group.

Bibliographie

[1] Hall P., Phillip Hall's lecture notes on group theory — Part 6, University of Cambridge, Cambridge, 1951–1967 URL: http://omeka.wustl.edu/omeka/items/show/10788

[2] de Giovanni F., Trombetti M., “Pronormality in group theory”, Adv. Group Theory Appl., 9 (2020), 123–149 | DOI | MR | Zbl

[3] Kondrat'ev A.S., Maslova N.V., Revin D.O., “On the pronormality of subgroups of odd index in finite simple groups”, Groups St Andrews 2017 in Birmingham, London Math. Soc. Lecture Note Ser., 455, eds. C.M. Campbell, M.R. Quick, C.W. Parker, E.F. Robertson, C. M. Roney-Dougal, Cambridge Univ. Press, Cambridge, 2019, 406–418 | DOI | MR | Zbl

[4] Guo W., Revin D.O., “Pronormality and submaximal $\mathfrak{X}$-subgroups in finite groups”, Comm. Math. Stat., 6:3 (2018), 289–317 | DOI | MR | Zbl

[5] Romalis G.M., Sesekin N.F., “O metagamiltonovykh gruppakh. I–III”, Mat. zap. UrGU, 5, tetr. 2 (1966), 45–49 ; 6, тетр. 5, 52–58; 7, тетр. 3 (1969/70), 195–199 | MR | MR

[6] De Falco M., de Giovanni F., Musella C., “Metahamiltonian groups and related topics”, Int. J. Group Theory, 2:1 (2013), 117–129 | MR | Zbl

[7] Brescia M., Ferrara M., Trombetti M., “The structure of metahamiltonian groups”, Jpn. J. Math., 18:1 (2023), 1–65 | DOI | MR | Zbl

[8] Makhnev A.A., “O konechnykh metagamiltonovykh gruppakh”, Mat. zap. UrGU, 10, tetr. 1 (1976), 60–75 | Zbl

[9] Brescia M., Ferrara M., Trombetti M., “Groups whose subgroups are either abelian or pronormal ”, Kyoto J. Math., 63:3 (2023), 471–500 | DOI | MR

[10] Brescia M., Trombetti M., “Locally finite simple groups whose non-Abelian subgroups are pronormal”, Comm. Algebra, 51:8 (2023), 3346–3353 | DOI | MR | Zbl

[11] Ferrara M., Trombetti M., “Groups with many pronormal subgroups”, Bull. Austral. Math. Soc., 105:1 (2022), 75–86 | DOI | MR | Zbl

[12] Ferrara M., Trombetti M., “Locally finite simple groups whose nonnilpotent subgroups are pronormal”, publ. online, Bull. Austral. Math. Soc., 2023, 1–10 | DOI

[13] Wielandt H., “Zusammengesetzte Gruppen endlicher Ordnung”, Lecture notes Math. Inst. Univ. Tübingen (1963/64), Helmut Wielandt. Mathematische Werke/Mathematical Works, v. 1, Group Theory, de Gruyter, Berlin, 1994, 607–655 | MR

[14] Go V., Revin D.O., “O maksimalnykh i submaksimalnykh $\mathfrak{X}$-podgruppakh”, Algebra i logika, 57:1 (2018), 14–42 | DOI | MR | Zbl

[15] Guo W., Revin D.O., “Classification and properties of the $\pi$-submaximal subgroups in minimal nonsolvable groups”, Bull. Math. Sci., 8:2 (2018), 325–351 | DOI | MR | Zbl

[16] Revin D.O., “Submaksimalnye razreshimye podgruppy nechetnogo indeksa v znakoperemennykh gruppakh”, Sib. mat. zhurn., 62:2 (2021), 387–401 | DOI | Zbl

[17] Vdovin E.P., Revin D.O., “Pronormalnost khollovykh podgrupp v konechnykh prostykh gruppakh”, Sib. mat. zhurn., 53:3 (2012), 527–542 | DOI | MR | Zbl

[18] Aschbacher M., “The subgroup structure of finite groups”, Finite simple groups: thirty years of the Atlas and beyond, Contemp. Math. AMS, 694, 2017, 111–121 | DOI | MR | Zbl

[19] Wilson R.A., “A negative answer to a question of Aschbacher”, Albanian J. Math., 12:1 (2018), 24–31 | DOI | MR | Zbl

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