Geodesic
On the heritability of the Sylow $\pi$-theorem by subgroups
Sbornik. Mathematics, Tome 211 (2020) no. 3, pp. 309-335 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Let $\pi$ be a set of primes. We say that the Sylow $\pi$-theorem holds for a finite group $G$, or $G$ is a $\mathscr D_\pi$-group, if the maximal $\pi$-subgroups of $G$ are conjugate. Obviously, the Sylow $\pi$-theorem implies the existence of $\pi$-Hall subgroups. In this paper, we give an affirmative answer to Problem 17.44, (b), in the Kourovka notebook: namely, we prove that in a $\mathscr D_\pi$-group an overgroup of a $\pi$-Hall subgroup is always a $\mathscr D_\pi$-group. Bibliography: 52 titles.
Keywords: finite group, $\pi$-Hall subgroup, group of Lie type, maximal subgroup.
Mots-clés : $\mathscr D_\pi$-group 
@article{SM_2020_211_3_a0,
     author = {E. P. Vdovin and N. Ch. Manzaeva and D. O. Revin},
     title = {On the heritability of the {Sylow} $\pi$-theorem by subgroups},
     journal = {Sbornik. Mathematics},
     pages = {309--335},
     year = {2020},
     volume = {211},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2020_211_3_a0/}
}
TY  - JOUR
AU  - E. P. Vdovin
AU  - N. Ch. Manzaeva
AU  - D. O. Revin
TI  - On the heritability of the Sylow $\pi$-theorem by subgroups
JO  - Sbornik. Mathematics
PY  - 2020
SP  - 309
EP  - 335
VL  - 211
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/SM_2020_211_3_a0/
LA  - en
ID  - SM_2020_211_3_a0
ER  - 
%0 Journal Article
%A E. P. Vdovin
%A N. Ch. Manzaeva
%A D. O. Revin
%T On the heritability of the Sylow $\pi$-theorem by subgroups
%J Sbornik. Mathematics
%D 2020
%P 309-335
%V 211
%N 3
%U http://geodesic.mathdoc.fr/item/SM_2020_211_3_a0/
%G en
%F SM_2020_211_3_a0
E. P. Vdovin; N. Ch. Manzaeva; D. O. Revin. On the heritability of the Sylow $\pi$-theorem by subgroups. Sbornik. Mathematics, Tome 211 (2020) no. 3, pp. 309-335. http://geodesic.mathdoc.fr/item/SM_2020_211_3_a0/

[1] Z. Arad, D. Chillag, “$\pi$-solvability and nilpotent Hall subgroups”, The Santa Cruz conference on finite groups (Univ. California, Santa Cruz, CA, 1979), Proc. Sympos. Pure Math., 37, Amer. Math. Soc., Providence, RI, 1980, 197–199 | DOI | MR | Zbl

[2] Z. Arad, D. Chillag, “Finite groups containing a nilpotent Hall subgroup of even order”, Houston J. Math., 7:1 (1981), 23–32 | MR | Zbl

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

[4] M. Aschbacher, R. Lyons, S. D. Smith, R. Solomon, The classification of finite simple groups. Groups of characteristic $2$ type, Math. Surveys Monogr., 172, Amer. Math. Soc., Providence, RI, 2011, xii+347 pp. | DOI | MR | Zbl

[5] A. Beltrán, M. J. Felipe, G. Malle, A. Moretó, G. Navarro, L. Sanus, R. Solomon, Pham Huu Tiep, “Nilpotent and abelian Hall subgroups in finite groups”, Trans. Amer. Math. Soc., 368:4 (2016), 2497–2513 | DOI | MR | Zbl

[6] A. Borel, R. Carter, C. W. Curtis, N. Iwahori, T. A. Springer, R. Steinberg, Seminar on algebraic groups and related finite groups (Institute for Advanced Study, Princeton, NJ, 1968/69), Lecture Notes in Math., 131, Springer-Verlag, Berlin–New York, 1970, viii+321 pp. | DOI | MR | Zbl

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

[8] S. A. Chunikhin, “O podgruppakh otnositelno razreshimykh grupp”, Dokl. AN SSSR, 58:7 (1947), 1295–1296 | MR | Zbl

[9] S. A. Chunikhin, “O $\Pi$-otdelimykh gruppakh”, Dokl. AN SSSR, 59:3 (1948), 443–445 | MR | Zbl

[10] S. A. Čunihin, On $\Pi$-properties of finite groups, Amer. Math. Soc. Translation, 1952, no. 72, Amer. Math. Soc., Providence, RI, 1952, 32 pp. | MR | MR | Zbl

[11] D. Gorenstein, R. Lyons, The local structure of finite groups of characteristic $2$ type, Mem. Amer. Math. Soc., 42, no. 276, Amer. Math. Soc., Providence, RI, 1983, vii+731 pp. | DOI | MR | Zbl

[12] D. Gorenstein, R. Lyons, R. Solomon, The classification of the finite simple groups. Number 3. Part I. Chapter A. Almost simple $K$-groups, Math. Surveys Monogr., 40.3, Amer. Math. Soc., Providence, RI, 1998, xvi+419 pp. | MR | Zbl

[13] F. Gross, “Odd order Hall subgroups of $\operatorname{GL}(n,q)$ and $\operatorname{Sp}(2n,q)$”, Math. Z., 187:2 (1984), 185–194 | DOI | MR | Zbl

[14] F. Gross, “On a conjecture of Philip Hall”, Proc. London Math. Soc. (3), 52:3 (1986), 464–494 | DOI | MR | Zbl

[15] F. Gross, “On the existence of Hall subgroups”, J. Algebra, 98:1 (1986), 1–13 | DOI | MR | Zbl

[16] F. Gross, “Conjugacy of odd order Hall subgroups”, Bull. London Math. Soc., 19:4 (1987), 311–319 | DOI | MR | Zbl

[17] F. Gross, “Odd order Hall subgroups of the classical linear groups”, Math. Z., 220 (1995), 317–336 | DOI | MR | Zbl

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

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

[20] P. Hall, “A note on soluble groups”, J. London Math. Soc., 3:2 (1928), 98–105 | DOI | MR | Zbl

[21] P. Hall, “A characteristic property of soluble groups”, J. London Math. Soc., 12:3 (1937), 198–200 | DOI | MR | Zbl

[22] P. Hall, “Theorems like Sylow's”, Proc. London Math. Soc. (3), 6:2 (1956), 286–304 | DOI | MR | Zbl

[23] I. M. Isaacs, Finite group theory, Grad. Stud. Math., 92, Amer. Math. Soc., Providence, RI, 2008, xii+350 pp. | DOI | MR | Zbl

[24] M. I. Kargapolov, Ju. I. Merzljakov, Fundamentals of the theory of groups, Grad. Texts in Math., 62, Springer-Verlag, New York–Berlin, 1979, xvii+203 pp. | MR | MR | Zbl | Zbl

[25] P. Kleidman, M. Liebeck, The subgroup structure of the finite classical groups, London Math. Soc. Lecture Note Ser., 129, Cambridge Univ. Press, Cambridge, 1990, x+303 pp. | DOI | MR | Zbl

[26] M. W. Liebeck, J. Saxl, G. M. Seitz, “Subgroups of maximal rank in finite exceptional groups of Lie type”, Proc. London Math. Soc. (3), 65:2 (1992), 297–325 | DOI | MR | Zbl

[27] M. W. Liebeck, G. M. Seitz, “Maximal subgroups of exceptional groups of Lie type, finite and algebraic”, Geom. Dedicata, 35:1-3 (1990), 353–387 | DOI | MR | Zbl

[28] G. Malle, “The maximal subgroups of ${}^2F_4(q^2)$”, J. Algebra, 139:1 (1991), 52–69 | DOI | MR | Zbl

[29] N. Ch. Manzaeva, “Reshenie problemy Vilanda dlya sporadicheskikh grupp”, Sib. elektron. matem. izv., 9 (2012), 294–305 | MR | Zbl

[30] N. Ch. Manzaeva, “Heritability of the property $\mathscr D_\pi$ by overgroups of $\pi$-Hall subgroups in the case where $2\in \pi$”, Algebra and Logic, 53:1 (2014), 17–28 | DOI | MR | Zbl

[31] A. Moretó, “Sylow numbers and nilpotent Hall subgroups”, J. Algebra, 379 (2013), 80–84 | DOI | MR | Zbl

[32] The Kourovka notebook. Unsolved problems in group theory, 19th ed., eds. V. D. Mazurov, E. I. Khukhro, Rus. Acad. Sci. Sib. Branch Inst. Math., Novosibirsk, 2018, 248 pp. | MR

[33] M. N. Nesterov, “On pronormality and strong pronormality of Hall subgroups”, Siberian Math. J., 58:1 (2017), 128–133 | DOI | DOI | MR | Zbl

[34] D. O. Revin, “The $\mathscr D_\pi$-property in finite simple groups”, Algebra and Logic, 47:3 (2008), 210–227 | DOI | MR | Zbl

[35] D. O. Revin, “Vokrug gipotezy F. Kholla”, Sib. elektron. matem. izv., 6 (2009), 366–380 | MR | Zbl

[36] 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

[37] R. Steinberg, “Automorphisms of finite linear groups”, Canad. J. Math., 12:4 (1960), 606–615 | DOI | MR | Zbl

[38] R. Steinberg, Endomorphisms of linear algebraic groups, Mem. Amer. Math. Soc., 80, Amer. Math. Soc., Providence, RI, 1968, 108 pp. | DOI | MR | Zbl

[39] M. L. Sylow, “Théorèmes sur les groupes de substitutions”, Math. Ann., 5:4 (1872), 584–594 | DOI | MR | Zbl

[40] E. P. Vdovin, N. Ch. Manzaeva, D. O. Revin, “On the heritability of the property $\mathscr D_\pi$ by subgroups”, Proc. Steklov Inst. Math., 279, suppl. 1 (2012), 130–138 | DOI | Zbl

[41] E. P. Vdovin, D. O. Revin, “Hall subgroups of odd order in finite groups”, Algebra and Logic, 41:1 (2002), 8–29 | DOI | MR | Zbl

[42] D. O. Revin, E. P. Vdovin, “Hall subgroups of finite groups”, Ischia group theory 2004, Contemp. Math., 402, Israel Math. Conf. Proc., Amer. Math. Soc., Providence, RI, 2006, 229–263 | DOI | MR | Zbl

[43] E. P. Vdovin, D. O. Revin, “Theorems of Sylow type”, Russian Math. Surveys, 66:5 (2011), 829–870 | DOI | DOI | MR | Zbl

[44] E. P. Vdovin, D. O. Revin, “Pronormality of Hall subgroups in finite simple groups”, Siberian Math. J., 53:3 (2012), 419–430 | DOI | MR | Zbl

[45] E. P. Vdovin, D. O. Revin, “On the pronormality of Hall subgroups”, Siberian Math. J., 54:1 (2013), 22–28 | DOI | MR | Zbl

[46] E. P. Vdovin, D. O. Revin, “The existence of pronormal $\pi$-Hall subgroups in $E_\pi$-groups”, Siberian Math. J., 56:3 (2015), 379–383 | DOI | DOI | MR | Zbl

[47] A. J. Weir, “Sylow $p$-subgroups of the classical groups over finite fields with characteristic prime to $p$”, Proc. Amer. Math. Soc., 6:4 (1955), 529–533 | DOI | MR | Zbl

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

[49] H. Wielandt, “Entwicklungslinien in der Strukturtheorie der endlichen Gruppen”, Proc. Internat. Congress Math. 1958, Cambridge Univ. Press, New York, 1960, 268–278 | MR | Zbl

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

[51] H. Wielandt, “Zusammengesetzte Gruppen endlicher Ordnung”, Vorlesung an der Universität Tübingen im Wintersemester 1963/64, Mathematische Werke/Mathematical works, v. 1, Group theory, Walter de Gruyter Co., Berlin, 1994, 607–656 | MR | Zbl

[52] H. Wielandt, “Zusammengesetzte Gruppen: Hölder Programm heute”, The Santa Cruz conference on finite groups (Univ. California, Santa Cruz, CA, 1979), Proc. Sympos. Pure Math., 37, Amer. Math. Soc., Providence, RI, 1980, 161–173 | DOI | MR | Zbl