Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-30T19:03:07.519Z Has data issue: false hasContentIssue false

Generalizations of the Sylow theorem

Published online by Cambridge University Press:  05 July 2011

By
Danila O. Revin  and
Evgeny P. Vdovin
Edited by
C. M. Campbell ,
M. R. Quick ,
E. F. Robertson ,
C. M. Roney-Dougal ,
G. C. Smith  and
G. Traustason
Danila O. Revin
Affiliation:
Novosibirsk State University
Evgeny P. Vdovin
Affiliation:
Novosibirsk State University
C. M. Campbell
Affiliation:
University of St Andrews, Scotland
M. R. Quick
Affiliation:
University of St Andrews, Scotland
E. F. Robertson
Affiliation:
University of St Andrews, Scotland
C. M. Roney-Dougal
Affiliation:
University of St Andrews, Scotland
G. C. Smith
Affiliation:
University of Bath
G. Traustason
Affiliation:
University of Bath
Get access

Summary

Abstract

Let π be a set of primes. Generalizing the properties of Sylow p-subgroups, P.Hall introduced classes Eπ, Cπ, and Dπ of finite groups possessing a π-Hall subgroup, possessing exactly one class of conjugate π-Hall subgroups, and possessing one class of conjugate maximal π-subgroups respectively. In this paper we discuss a description of these classes in terms of a composition and a chief series of a finite group G.

Introduction

In 1872, the Norwegian mathematician L. Sylow proved the following outstanding theorem.

Theorem 1.1 (L. Sylow [76])Let G be a finite group and p a prime. Assume |G| = pαm and (p, m) = 1. Then the following statements hold:

  1. (E) G possesses a subgroup of order pα (the, so-called, Sylow p-subgroup);

  2. (C) every two Sylow p-subgroups of G are conjugate;

  3. (D) every p-subgroup of G is included in a Sylow p-subgroup.

A natural generalization of the concept of Sylow p-subgroups is the notion of π-Hall subgroups. We recall the definitions. Let G be a finite group and π be a set of primes. We denote by π′ the set of all primes not in π, by π(n) the set of all prime divisors of a positive integer n and for a finite group G we denote π(|G|) by π(G). A positive integer n with π(n) ⊆ π is called a π-number and a group G with π(G) ⊆ π is called a π-group.

Type
Chapter
Information
Groups St Andrews 2009 in Bath , pp. 488 - 519
Publisher: Cambridge University Press
Print publication year: 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Z., Arad and M. B., Ward, New criteria for the solvability of finite groups, J. Algebra 77 (1982), N1, 234–246.Google Scholar
[2] Z., Arad and D., Chilag, A criterion for the existence of normal π-complements in finite groups, J. Algebra 87 (1984), N2, 472–482.Google Scholar
[3] Z., Arad and E., Fisman, On finite factorizable groups, J. Algebra 86 (1984), N2, 522–548.Google Scholar
[4] R., Baer, Verstreute Untergruppen endlicher Gruppen, Arch. Math. 9 (1958), N1–2, 7–17.Google Scholar
[5] R. W., Carter, Simple groups of Lie type, John Wiley and Sons, London, 1972.Google Scholar
[6] P., Cobb, Existence of Hall subgroups and embedding of π-subgroups into Hall subgroups, J. Algebra 127 (1989), N1, 229–243.Google Scholar
[7] J. H., Conway, R. T., Curtis, S. P., Norton, R. A., Parker and R. A., Wilson, Atlas of Finite Groups (OUP, Oxford 1985)Google Scholar
[8] M. P., Coone, The nonsolvable Hall subgroups of the general linear groups, Math. Z. 114 (1970), N4, 245–270.Google Scholar
[9] S. A., Chunikhin, On solvable groups, Izv. NIIMM Tom. Univ. 2 (1938), 220–223 (in Russian).Google Scholar
[10] S. A., Chunikhin, On Sylow-regular groups, Doklady Akad. Nauk SSSR 60 (1948), N5, 773–774 (in Russian).Google Scholar
[11] S. A., Chunikhin, On Π-properties of finite groups, Amer. Math. Soc. Translation (1952). no. 72. Translated from Mat. Sbornik 25 (1949), N3, 321–346.Google Scholar
[12] S. A., Chunikhin, On the conditions of theorems of Sylow's type, Doklady Akad. Nauk SSSR 69 (1949), N6, 735–737 (in Russian).Google Scholar
[13] S. A., Chunikhin, On Sylow properties of finite groups, Doklady Akad. Nauk SSSR 73 (1950), N1, 29–32 (in Russian).Google Scholar
[14] S. A., Chunikhin, On weakening the conditions in theorems of Sylow type, Doklady Akad. Nauk SSSR 83 (1952), N5, 663–665 (in Russian).Google Scholar
[15] S. A., Chunikhin, On subgroups of a finite group, Doklady Akad. Nauk SSSR 86 (1952), N1, 27–30 (in Russian).Google Scholar
[16] S. A., Chunikhin, On existence and conjugateness of subgroups of a finite group, Mat. Sbornik 33 (1953), N1, 111–132 (in Russian).Google Scholar
[17] S. A., Chunikhin, On Π-solvable subgroups of finite groups, Doklady Akad. Nauk SSSR 103 (1955), N3, 377–378 (in Russian).Google Scholar
[18] S. A., Chunikhin, Some trends in the development of the theory of finite groups in recent years, Uspehi Mat. Nauk 16 (1961), N4 (100), 31–50 (in Russian).Google Scholar
[19] S. A., Chunikhin, On indexials of finite groups, Soviet Math. Dokl. 121 (1961), N4 (100), 70–71. Translated from the Russian Doklady Akad. Nauk SSSR136, 299–300.Google Scholar
[20] S. A., Chunikhin, A π-Sylow theorem following from the hypothesis of solvability of groups of odd order, Dokl. Akad. Nauk BSSR 6 (1962), N6, 345–346 (in Russian).Google Scholar
[21] S. A., Chunikhin, Subgroups of finite groups. Wolters-Noordhoff Publishing, Groningen 1969.Google Scholar
[22] S. A., Chunikhin and L. A., Shemetkov, Finite groups, J. Soviet Math. 1 (1973), no. 3., Plenum Press, New York, 1973. 291–390, Translated from the Russian Algebra. Topology. Geometry, 1969, 7–70, Akad. Nauk SSSR Vsesojuz. Inst. Nauchn. i Tehn. Informacii, Moscow, 1971.Google Scholar
[23] S. A., Chunikhin, On the definition of π-solvable and π-separable groups, Dokl. Akad. Nauk SSSR 212 (1973), 1078–1081 (in Russian).Google Scholar
[24] K., Doerk and T., Hawks, Finite soluble groups, Berlin, New York, Walter de Gruyter, 1992.Google Scholar
[25] W., Feit and J. G., Thompson, Solvability of groups of odd order, Pacif. J. Math. 13 (1963) N3, 775–1029.Google Scholar
[26] P. A., Ferguson and P., Kelley, Hall π-subgroups which are direct product of nonabelian simple groups, J. Algebra 120 N1 (1989), 40–46.Google Scholar
[27] F., Gross, Odd order Hall subgrous of GL(n, q) and Sp(2n, q), Math. Z. 187 (1984), N2, 185–194.Google Scholar
[28] F., Gross, On the existence of Hall subgroups, J. Algebra 98 (1986), N1, 1–13.Google Scholar
[29] F., Gross, On a conjecture of Philip Hall, Proc. London Math. Soc. Ser. III 52 (1986), N3, 464–494.Google Scholar
[30] F., Gross, Odd order Hall subgroups of the classical linear groups, Math. Z. 220 (1995), N3, 317–336.Google Scholar
[31] F., Gross, Conjugacy of odd order Hall subgroups, Bull. London Math. Soc. 19 (1987), N4, 311–319.Google Scholar
[32] F., Gross, Hall subgroups of order not divisible by 3, Rocky Mt. J. Math. 23 (1993), N2, 569–591.Google Scholar
[33] W., Guo and B., Li, On the Shemetkov problem for Fitting classes, Beiträge Algebra Geom. 48 (2007), N1, 281–289.Google Scholar
[34] W., Guo, Formations determined by Hall subgroups, J. Appl. Algebra Discrete Struct. 4 (2006), N3, 139–147.Google Scholar
[35] W., Guo, Some problems and results for the research on Sylow objects of finite groups. J. Xuzhou Norm. Univ. Nat. Sci. Ed. 23 (2005), N3, 1–6, 40 (in Chinese).Google Scholar
[36] P., Hall, Theorems like Sylow's, Proc. London Math. Soc. (3) 6 (1956), 286–304.Google Scholar
[37] P., Hall, A note on soluble groupsJ. London Math. Soc. 3 (1928), 98–105.Google Scholar
[38] P., Hall, A characteristic property of soluble groups, J. London Math. Soc. 12 (1937), 198–200.Google Scholar
[39] P., Hall, On the Sylow system of a soluble group, Proc. London Math. Soc. Ser. II 43 (1937), 316–320.Google Scholar
[40] B., Hartley, A theorem of Sylow type for a finite groups, Math. Z. 122 (1971), N4, 223–226.Google Scholar
[41] B., Hartley, Helmut Wielandt on the π-structure of finite groups, Mathematische Werke = Mathematical Works / Helmut Wielandt, ed. by B., Huppert and H., Sneider, vol. 1, Walter de Gruyter, Berlin, 1994, 511–516.Google Scholar
[42] J. E., Humphreys, Linear algebraic groups, Springer-Verlag, New York, 1972.Google Scholar
[43] N., ItoOn π-structures of finite groups, Tôhoku Math. Journ. 4 (1952) N1, 172–177.Google Scholar
[44] P. B., Kleidman and M., Liebeck, The subgroups structure of finite classical groups, Cambridge Univ. Press, 1990.Google Scholar
[45] L. S., Kazarin, Sylow type theorems for finite groups, Sructure properties of algebraic systems, Nalchik, Kabardino-Balkarskii Univ., 1981, 42–52 (in Russian).Google Scholar
[46] L. S., Kazarin, On the product of finite groups, Doklady Akad. Nauk SSSR 269 (1983), N3, 528–531 (in Russian).Google Scholar
[47] M. I., Kargapolov, Factorization of Π-separable groups, Doklady Akad. Nauk SSSR 114 (1957), N6, 1155–1157 (in Russian).Google Scholar
[48] A. S., Kondrat'ev, Subgroups of finite Chevalley groups, Russian Math. Surveys 41, N1 (1986), 65–118.Google Scholar
[49] V. D., Mazurov, On a question of L. A. Shemetkov, Algebra and Logic 31 (1992), N6, 360–366.Google Scholar
[50] V. D., Mazurov and D. O., Revin, On the Hall Dπ-property for finite groups, Siberian Math. J. 38 (1997), N1, 106–113.Google Scholar
[51] D. O., Revin, Hall π-subgroups of finite Chevalley groups whose characteristic belongs to π, Siberian Adv. Math. 9 (1999), N2, 25–71.Google Scholar
[52] D. O., Revin, The Dπ-property in a class of finite groups, Algebra and Logic 41 (2002) N3, 187–206.Google Scholar
[53] D. O., Revin and E. P., Vdovin, Hall subgroups of finite groups, Contemporary Mathematics 402 (2006), 229–265.Google Scholar
[54] D. O., Revin, The Dπ property of finite groups in the case 2 ∉ π, Proc. Steklov Inst. Math. 257 (2006) Suppl. 1, S164–S180.Google Scholar
[55] D. O., Revin, A characterization of finite Dπ-groups, Doklady Math. 76 (2007), N3, 925–928.Google Scholar
[56] D. O., Revin, The property Dπ in linear and unitary groups, Sib. Math. J. 49 (2008), N2, 353–361.Google Scholar
[57] D. O., Revin, The Dπ-property in finite simple groups, Algebra and Logic 47 (2008), N3, 210–227.Google Scholar
[58] D. O., Revin and E. P., Vdovin, On the number of classes of conjugate Hall subgroups in finite simple groups, in preparation.
[59] D. O., Revin and E. P., Vdovin, Existence criterion for Hall subgroups of finite groups, in preparation.
[60] D. O., Revin and E. P., Vdovin, Conjugacy criterion for Hall subgroups of finite groups, Siberian Math. J., to appear.
[61] S. A., Rusakov, Analogues of Sylow's theorem on inclusion of subgroups, Doklady Akad. Nauk SSSR 5 (1961), 139–141 (in Russian).Google Scholar
[62] S. A., Rusakov, Analogues of Sylow's theorem on the existence and imbedding of subgroups, Sibirsk. Mat. Zh. 4 (1963), N5, 325–342 (in Russian).Google Scholar
[63] S. A., Rusakov, C-theorems for n-groups, Vesci Akad. Navuk BSSR Ser. Fiz.-Mat. Navuk, 1972, N3, 5–9 (in Russian).Google Scholar
[64] L. A., Shemetkov, On a theorem of Hall, Dokl. Akad. Nauk SSSR 147 (1962), 321–322 (in Russian).Google Scholar
[65] L. A., Shemetkov, D-structure of finite groups, Mat. Sb. 67 (1965), N3, 384–497 (in Russian).Google Scholar
[66] L. A., Shemetkov, A new D-theorem in the theory of finite groups, Dokl. Akad. Nauk SSSR 160 (1965), N2, 290–293 (in Russian).Google Scholar
[67] L. A., Shemetkov, Sylow properties of finite groups, Mat. Sb. 76 (1968), N2, 271–287 (in Russian).Google Scholar
[68] L. A., Shemetkov, Conjugacy and imbedding of subgroups, Finite Groups, Nauka i Tehnika, Minsk, 1966, 881–883 (in Russian).Google Scholar
[69] L. A., Shemetkov, Sylow properties of finite groups, Dokl. Akad. Nauk BSSR 16 (1972), N10, 881–883.Google Scholar
[70] L. A., Shemetkov, Two trends in the development of the theory of nonsimple finite groups, Uspehi Mat. Nauk 30 (1975), N2(182), 179–198 (in Russian).Google Scholar
[71] L. A., Shemetkov, Formatsii konechnykh grupp (Formations of finite groups), Monographs in Modern Algebra, (“Nauka”, Moscow, 1978) (in Russian).Google Scholar
[72] L. A., Shemetkov and A. F., Vasil'ev, Nonlocal formations of finite groupsDokl. Akad. Nauk Belarusi 39 (1995), 5–8 (in Russian).Google Scholar
[73] L. A., Shemetkov, A generalization of Sylow's theorem, Siberian Math. J. 44 (2003), N6, 1127–1132.Google Scholar
[74] E. L., Spitznagel, Hall subgroups of certain families of finite groups, Math. Z. 97 (1967), N4, 259–290.Google Scholar
[75] M., Suzuki, Group theory II, NY, Berlin, Heidelberg, Tokyo, Springer-Verl. 1986.Google Scholar
[76] M. L., Sylow, Théorèmes sur les groupes de substitutions, Math. Ann. 5 (1872), N4, 584–594.Google Scholar
[77] M. C., Tibiletti, Sui prodotti ordinati di gruppi finiti, Boll. Un. Mat. Ital. (3) 13 (1958), 46-57.Google Scholar
[78] J. G., Thompson, Hall subgroups of the symmetric groups, J. Comb. Th. 1 (1966) N2, 271–279.Google Scholar
[79] V. N., Tyutyanov, The Dπ-theorem for finite groups having composition factors such that the 2-length of any solvable subgroup does not exceed one, Vestsi Nats. Akad. Navuk Belarusi Ser. Fiz.-Mat. Navuk, 2000 N1, 12–14 (in Russian).Google Scholar
[80] V. N., Tyutyanov, Sylow-type theorems for finite groups, Ukrainian Math. J. 52 (2000), N10, 1628–1633.Google Scholar
[81] V. N., Tyutyanov, On the Hall conjecture, Ukrainian Math. J. 54 (2002), N7, 1181–1191.Google Scholar
[82] E. P., Vdovin and D. O., Revin, Hall subgroups of odd order in finite groups, Algebra and Logic 41 (2002) N1, 8–29.Google Scholar
[83] H., Wielandt, Zum Satz von Sylow, Math. Z. 60 (1954), N4. 407–408.Google Scholar
[84] H., Wielandt, Sylowgruppen und Kompositoin-Struktur, Abh. Math. Sem. Univ. Hamburg 22 (1958), 215–228.Google Scholar
[85] H., Wielandt, Zum Satz von Sylow. II, Math. Z. 71 (1959), N4. 461–462.Google Scholar
[86] H., Wielandt, Entwicklungslinien in der Strukturtheorie der endlichen Gruppen, Proc. Intern. Congress Math., Edinburg, 1958. London: Cambridge Univ. Press, 1960, 268–278.Google Scholar
[87] H., Wielandt, Arithmetische Struktur und Normalstuktur endlicher Gruppen, Conv. Internaz. di Teoria dei Gruppi Finitine Applicazioni, Firenze, 1960. Roma: Edizioni Cremonese, 1960, 56–65.Google Scholar
[88] H., Wielandt, Der Normalisator einer subnormalen Untergruppe, Acta Sci. Math. Szeged 21 (1960) 324–336.Google Scholar
[89] H., Wielandt, Sylowtürme in subnormalen Untergruppen, Math. Z. 73 (1960), N4. 386–392.Google Scholar
[90] H., Wielandt and B., Huppert, Arithmetical and normal sructureof finite groups, Proc. Symp. Pure Math. 6 (1962), Providence RI: Amer. Math. Soc., 17–38.Google Scholar
[91] H., Wielandt, Sur la Stucture des groupes composés, Séminare Dubriel-Pisot(Algèbre et Théorie des Nombres), 17e anée, 10 pp. 1963/64. N17.Google Scholar
[92] H., Wielandt, Zusammenghesetzte Gruppen: Hölder Programm heute, The Santa Cruz conf. on finite groups, Santa Cruz, 1979. Proc. Sympos. Pure Math. 37, Providence RI: Amer. Math. Soc., 1980, 161–173.Google Scholar
[93] G., Zappa, Sopra unestensione di Wielandt del teorema di Sylow, Boll. Un. Mat. Ital. (3) 9, (1954), N4, 349–353.Google Scholar
[94] H., Zassenhaus, Lehrbuch der Gruppentheorie, Leipzig, Berlin. 1937.Google Scholar
[95] The Kourovka notebook. Unsolved problems in group theory. Edited by V. D. Mazurov and E. I. Khukhro. 16-th. ed., Russian Academy of Sciences Siberian Division, Institute of Mathematics, Novosibirsk, 2006.

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×