Maximal and submaximal $\mathfrak X$-subgroups

W. Guo ; D. O. Revin

Algebra i logika, Tome 57 (2018) no. 1, pp. 14-42.

Résumé

Let $\mathfrak X$ be a class of finite groups closed under taking subgroups, homomorphic images, and extensions. Following H. Wielandt, we call a subgroup $H$ of a finite group $G$ a submaximal $\mathfrak X$-subgroup if there exists an isomorphic embedding $\phi\colon G\hookrightarrow G^*$ of $G$ into some finite group $G^*$ under which $G^\phi$ is subnormal in $G^*$ and $H^\phi=K\cap G^\phi$ for some maximal $\mathfrak X$-subgroup $K$ of $G^*$. In the case where $\mathfrak X$ coincides with the class of all $\pi$-groups for some set $\pi$ of prime numbers, submaximal $\mathfrak X$-subgroups are called submaximal $\pi$-subgroups. In his talk at the well-known conference on finite groups in Santa Cruz in 1979, Wielandt emphasized the importance of studying submaximal $\pi$-subgroups, listed (without proof) certain of their properties, and formulated a number of open questions regarding these subgroups. Here we prove properties of maximal and submaximal $\mathfrak X$- and $\pi$-subgroups and discuss some open questions both Wielandt’s and new ones. One of such questions due to Wielandt reads as follows: Is it always the case that all submaximal $\mathfrak X$-subgroups are conjugate in a finite group $G$ in which all maximal $\mathfrak X$-subgroups are conjugate?

Détail

Keywords: finite group, maximal $\mathfrak X$-subgroup, submaximal $\mathfrak X$-subgroup, Hall $\pi$-subgroup, $\mathscr D_\pi$-property.

Bibliographie

[1] H. Wielandt, “Zusammengesetzte Gruppen: Holders Programm heute”, Finite groups (Santa Cruz Conf., 1979), Proc. Symp. Pure Math., 37, Am. Math. Soc., Providence, RI, 1980, 161–173 | DOI | MR | Zbl

[2] H. Wielandt, Sur la Stucture des groupes composés, Séminare Dubriel-Pisot, Algèbre et Théorie des Nombres, 17e anée, No. 17, 1963/64, 10 pp.

[3] H. Wielandt, “Zusammengesetzte Gruppen endlicher Ordnung”, Vorlesung an der Univ. Tübingen im Wintersemester 1963/64: H. Wielandt, Mathematische Werke. Mathematical works, v. 1, Group theory, eds. B. Huppert, H. Schneider, Walter de Gruyter, Berlin, 1994, 607–516 | MR

[4] P. Hall, “Theorems like Sylow's”, Proc. Lond. Math. Soc. III Ser., 6 (1956), 286–304 | DOI | MR | Zbl

[5] P. Hall, “A note on soluble groups”, J. Lond. Math. Soc., 3 (1928), 98–105 | DOI | MR | Zbl

[6] P. Hall, “A characteristic property of soluble groups”, J. Lond. Math. Soc., 12 (1937), 198–200 | DOI | MR | Zbl

[7] S. A. Chunikhin, “O razreshimykh gruppakh”, Izv. NIIMM Tom. un-ta, 1938, no. 2, 220–223 | Zbl

[8] H. Wielandt, “Entwicklungslinien in der Strukturtheorie der endlichen Gruppen”, Proc. Int. Congr. Math. (Edinburgh, 14–21 Aug. 1958), Cambridge Univ. Press, New York, 1960, 268–278 | MR

[9] H. Wielandt, “Zum Satz von Sylow”, Math. Z., 60:4 (1954), 407–408 | DOI | MR | Zbl

[10] H. Wielandt, “Sylowgruppen und Kompositions-Struktur”, Abh. Math. Semin. Univ. Hamb., 22 (1958), 215–228 | DOI | MR | Zbl

[11] H. Wielandt, “Zum Satz von Sylow. II”, Math. Z., 71:4 (1959), 461–462 | DOI | MR | Zbl

[12] H. Wielandt, “Arithmetische Struktur und Normalstruktur endlicher Gruppen”, Convegno internaz. Teoria Gruppi finiti Appl. (Univ. Firenze, 1960), Edizioni Cremonese, Roma, 1960, 56–65 | MR

[13] H. Wielandt, “Sylowtürme in subnormalen Untergruppen”, Math. Z., 73:4 (1960), 386–392 | DOI | MR | Zbl

[14] H. Wielandt, B. Huppert, “Arithmetical and normal structure of finite groups”, Proc. Sympos. Pure Math., 6, Am. Math. Soc., Providence, RI, 1962, 17–38 | DOI | MR | Zbl

[15] B. Hartley, “Helmut Wielandt on the $\pi$-structure of finite groups”: H. Wielandt, Mathematische Werke. Mathematical works, v. 1, Group theory, eds. B. Huppert, H. Schneider, Walter de Gruyter, Berlin, 1994, 511–516 | MR

[16] E. P. Vdovin, D. O. Revin, “Teoremy silovskogo tipa”, Uspekhi matem. n., 66:5(401) (2011), 3–46 | DOI | MR | Zbl

[17] D. O. Revin, E. P. Vdovin, “An existence criterion for Hall subgroups of finite groups”, J. Group Theory, 14:1 (2011), 93–101 | DOI | MR | Zbl

[18] W. Guo, Structure theory of canonical classes of finite groups, Springer-Verlag, Berlin, 2015 | MR | Zbl

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

[20] D. O. Revin, E. P. Vdovin, “On the number of classes of conjugate Hall subgroups in finite simple groups”, J. Algebra, 324:12 (2010), 3614–3652 | DOI | MR | Zbl

[21] W. Guo, The theory of classes of groups, Math. Appl. (Dordrecht), 505, Kluwer Acad. Publ., Dordrecht; Sci. Press, Beijing, 2000 | MR | Zbl

[22] L. A. Shemetkov, Formatsii konechnykh grupp, Nauka, M., 1978 | MR

[23] M. Suzuki, Group Theory, v. II, Grundlehren Mathem. Wiss., 248, Springer-Verlag, New York etc., 1986 | DOI | MR | Zbl

[24] Nereshënnye voprosy teorii grupp. Kourovskaya tetrad, 18-e izd., In-t matem. SO RAN, Novosibirsk, 2014 http://www.math.nsc.ru/~alglog/alglog.html

[25] D. O. Revin, E. P. Vdovin, “Hall subgroups of finite groups”, Ischia group theory 2004, Proc. conf. honor Marcel Herzog (Naples, Italy, March 31 – April 03, 2004), Contemp. Math., 402, Israel Math. Conf. Proc., eds. Arad, Zvi et al., Am. Math. Soc., Providence, RI; Bar-Ilan Univ., Ramat Gan, 2006, 229–263 | DOI | MR | Zbl

[26] D. O. Revin, “Svoistvo $D_\pi$ v konechnykh prostykh gruppakh”, Algebra i logika, 47:3 (2008), 364–394 | MR | Zbl

[27] W. Guo, D. O. Revin, E. P. Vdovin, “Confirmation for Wielandt's conjecture”, J. Algebra, 434 (2015), 193–206 | DOI | MR | Zbl

[28] B. Hartley, “A theorem of Sylow type for finite groups”, Math. Z., 122:4 (1971), 223–226 | DOI | MR | Zbl

[29] L. A. Shemetkov, “Obobscheniya teoremy Silova”, Sib. matem. zh., 44:6 (2003), 1425–1431 | MR | Zbl

[30] V. Go, D. O. Revin, “O svyazi mezhdu sopryazhënnostyu maksimalnykh i submaksimalnykh $\mathfrak X$-podgrupp”, Algebra i logika (to appear)

[31] N. V. Maslova, D. O. Revin, “O neabelevykh kompozitsionnykh faktorakh konechnoi gruppy, minimalnoi otnositelno prostogo spektra”, Tr. IMM UrO RAN, 19, no. 4, 2013, 155–166 | MR

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

[33] J. Humphreys, Modular representations of finite groups of Lie type, London Math. Soc. Lect. Note Series, 326, Cambgidge Univ. Press, Cambgridge, 2006 | MR | Zbl

[34] M. W. Liebeck, J. Saxl, “The primitive permutation groups of odd degree”, J. Lond. Math. Soc. II Ser., 31:2 (1985), 250–264 | DOI | MR | Zbl

[35] W. M. Kantor, “Primitive permutation groups of odd degree, and an application to finite projective planes”, J. Algebra, 106:1 (1987), 15–45 | DOI | MR | Zbl

[36] N. V. Maslova, “Klassifikatsiya maksimalnykh podgrupp nechetnogo indeksa v konechnykh prostykh klassicheskikh gruppakh”, Tr. In-ta matem. mekh. UrO RAN, 14, no. 4, 2008, 100–118

[37] N. V. Maslova, “Klassifikatsiya maksimalnykh podgrupp nechetnogo indeksa v konechnykh gruppakh so znakoperemennym tsokolem”, Tr. In-ta matem. mekh. UrO RAN, 16, no. 3, 2010, 182–184

[38] N. V. Maslova, “Maksimalnye podgruppy nechetnogo indeksa v konechnykh gruppakh s prostym ortogonalnym tsokolem”, Tr. In-ta matem. mekh. UrO RAN, 16, no. 4, 2010, 237–245

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

[40] N. V. Maslova, “Classification of maximal subgroups of odd index in finite simple classical groups: Addendum”, Sib. elektron. matem. izv. (to appear)

[41] C. D. H. Cooper, “Maximal $\pi$-subgroups of the symmetric groups”, Math. Z., 123 (1971), 285–289 | DOI | MR | Zbl

[42] A. S. Kondratev, N. V. Maslova, D. O. Revin, “Kriterii pronormalnosti dobavlenii k abelevym normalnym podgruppam”, Tr. In-ta matematiki i mekhaniki UrO RAN, 22, no. 1, 2016, 153–158 | MR

[43] A. S. Kondratev, N. V. Maslova, D. O. Revin, “O pronormalnosti podgrupp nechetnogo indeksa v konechnykh prostykh gruppakh”, Sib. matem. zh., 56:6 (2015), 1375–1383 | DOI | MR | Zbl

[44] A. S. Kondratev, N. V. Maslova, D. O. Revin, “O pronormalnosti podgrupp nechetnykh indeksov v konechnykh prostykh simplekticheskikh gruppakh”, Sib. matem. zh., 58:3 (2017), 599–610 | DOI | MR | Zbl

[45] P. B. Kleidman, “A proof of the Kegel–Wielandt conjecture on subnormal subgroups”, Ann. Math. (2), 133:2 (1991), 369–428 | DOI | MR | Zbl

[46] N. Ch. Manzaeva, “Reshenie problemy Vilanda dlya sporadicheskikh grupp”, Sib. elektron. matem. izv., 9 (2012), 294–305 http://semr.math.nsc.ru/v9/p294-305.pdf | MR | Zbl

[47] D. O. Revin, “Vokrug gipotezy F. Kholla”, Sib. elektron. matem. izv., 6 (2009), 366–380 http://semr.math.nsc.ru/v9/p366-380.pdf | MR | Zbl

[48] E. P. Vdovin, N. Ch. Manzaeva, D. O. Revin, “O nasleduemosti svoistva $\mathcal D_\pi$ podgruppami”, Tr. In-ta matem. mekh. UrO RAN, 17, no. 4, 2011, 44–52

[49] W. Guo, D. O. Revin, “Classification and properties of the $\pi$-submaximal subgroups in minimal nonsolvable groups”, Bull. Math. Sci., 8:2 (2018), 325–351 ; arXiv: 1706.02016[math.GR] | DOI | MR

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

[51] H. Wielandt, “Eine Verallgemeinerung der invarianten Untergruppen”, Math. Z., 45 (1939), 209–244 | DOI | MR | Zbl