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Pronormality and Submaximal \(\mathfrak {X}\)-Subgroups on Finite Groups

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Abstract

According to Hall, a subgroup H of a group G is said to be pronormal if H and \(H^g\) are conjugate in \(\langle H,H^g\rangle \) for every \(g\in G\). In this survey, we discuss the role of pronormality for some subgroups of finite groups: Hall subgroups, subgroups of odd index, submaximal \(\mathfrak X\)-subgroup, etc.

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Notes

  1. Recall that a normal subset D of a group G is a set of odd transpositions (resp., 3-transpositions) if \(|t|=2\) and |st| either equals 2 or is an odd number (resp., \(|st|\in \{1,2,3\}\)) for every \(s,t\in D\).

  2. Recall that if K is a subgroup of G, then an overgroup of K is a subgroup of G containing K.

  3. Recall that G is said to be almost simple if its socle is a non-Abelian simple group. In other words, G is isomorphic to a subgroup of Aut(S) that contains Inn(S) for some non-Abelian simple group S.

  4. Recall that every \(\langle \textsc {s}_n,\textsc {n}_0\rangle \)-closed class of groups is called a Fitting class.

  5. First inclusion follows from the Sylow theorem and from the solvability of groups of prime power order. The second one is obvious.

  6. Wielandt’s proof can be found in his lectures [71, 13.2].

  7. Wielandt refers to a subgroup H of a group G as intravariant if its conjugacy class in G is invariant under the natural action of the group \({{\mathrm{Aut}}}(G)\) on the conjugacy classes of subgroups.

References

  1. Arad, Z., Ward, M.B.: New criteria for the solvability of finite groups. J. Algebra 77(1), 234–246 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aschbacher, M.: 3-Transposition Groups. Cambridge Tracts in Mathematics, vol. 124. Cambridge University Press, Cambridge (1997)

    Google Scholar 

  3. Aschbacher, M., Hall, M., Hall Jr., M.: Groups generated by a class of elements of order 3. J. Algebra 24, 591–612 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  4. Babai, L.: Isomorphism problem for a class of point-symmetric structures. Acta Math. Acad. Sci. Hungar. 29, 329–336 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  5. Buturlakin, A.A., Revin, D.O.: On \(p\)-complements of finite groups. Siberian Electron. Math. Rep. 10, 414–417 (2013)

    MathSciNet  MATH  Google Scholar 

  6. Cooper, C.D.H.: Maximal \(\pi \)-subgroups of the symmetric groups. Math. Z. 124(N4), 285–289 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chunikhin, S. A.: Über auflösbare Gruppen, Mitt. Forsch.-Inst. Math. Mech. Univ. Tomsk, 2 222–223 (1938)

  8. Doerk, K., Hawks, T.: Finite Soluble Groups. Walter de Gruyter, Berlin (1992)

    Book  Google Scholar 

  9. Gross, F.: Conjugacy of odd order Hall subgroups. Bull. Lond. Math. Soc. 19(4), 311–319 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  10. Guo, W.: The Theory of Classes of Groups. Kluwer Acad. Publ, Beijing (2006)

    Google Scholar 

  11. Guo, W.: Structure Theory of Canonical Classes of Finite Groups. Springer, Berlin (2015)

    Book  MATH  Google Scholar 

  12. Guo, W., Buturlakin, A.A., Revin, D.O.: Equivalency of the existence of non-conjugate and non-isomorphic Hall \(\pi \)-subgroups. Proc. Steklov Inst., accepted

  13. Guo, W., Maslova, N.V., Revin, D.O.: On the pronormality of subgroups of odd index in some extensions of finite groups. Siberian Math. J. 59(4), 610–622 (2018)

    Article  Google Scholar 

  14. Guo, W., Revin, D.O.: On the class of groups with pronormal Hall \(\pi \)-subgroups. Siberian Math. J. 55(3), 415–427 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Guo, W., Revin, D.O.: Classification and properties of the \(\pi \)-submaximal subgroups in minimal nonsolvable groups. Bull. Math. Sci. 8(2), 325–351 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  16. Guo, W., Revin, D.O.: On maximal and submaximal \({\mathfrak{X}}\)-subgroups. Algebra Logic 57(1), 9–28 (2018)

    Article  Google Scholar 

  17. Guo, W., Revin, D.O.: On a relation between the conjugateness for the maximal and submaximal \({\mathfrak{X}}\)-subgroups. Algebra Logic, accepted

  18. Guo, W., Revin, D.O., Vdovin, E.P.: Confirmation for Wielandt’s conjecture. J. Algebra 434, 193–206 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Guo, W., Revin, D.O., Vdovin, E.P.: Finite groups in which the \({\mathfrak{X}}\)-maximal subgroups are conjugate, in preparation

  20. Hartley, B.: A theorem of Sylow type for a finite groups. Math. Z. 122(4), 223–226 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hartley, B.: Helmut Wielandt on the \(\pi \)-structure of finite groups, Helmut Wielandt: Mathematical Works, Vol. 1, Group theory (ed. B. Huppert and H. Schneider, de Gruyter, Berlin, 1994), 511–516 (1994)

  22. Hall, P.: A note on soluble groups. J. Lond. Math. Soc. 3, 98–105 (1928)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hall, P.: A characteristic property of soluble groups. J. Lond. Math. Soc. 12, 198–200 (1937)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hall, P.: Theorems like Sylow‘s. Proc. Lond. Math. Soc. 6(22), 286–304 (1956)

    Article  MathSciNet  MATH  Google Scholar 

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

  26. Kantor, W.M.: Primitive permutation groups of odd degree, and an application to finite projective planes. J. Algebra. 106(1), 15–45 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  27. 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. London Mathematical Society Lecture Note Series (to appear). arXiv:1807.00384

  28. Kondrat’ev, A.S., Maslova, N.V., Revin, D.O.: On the pronormality of subgroups of odd index in finite simple groups. Siberian Math. J. 56(6), 1001–1007 (2015)

    MathSciNet  MATH  Google Scholar 

  29. Kondrat’ev, A.S., Maslova, N.V., Revin, D.O.: A pronormality criterion for supplements to Abelian normal subgroups. Proc. Steklov Inst. Math. 296(Suppl. 1), 1145–1150 (2017)

    MathSciNet  MATH  Google Scholar 

  30. Kondrat’ev, A.S., Maslova, N.V., Revin, D.O.: On the pronormality of subgroups of odd index in finite simple symplectic groups. Siberian Math. J. 58(3), 467–475 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kondrat’ev, A.S., Maslova, N.V., Revin, D.O.: On pronormal subgroups in finite simple groups. Doklady Math., 98(2), 1–4 (2018). https://doi.org/10.1134S1064562418060029

  32. Kondrat’ev, A.S., Maslova, N.V., Revin, D.O.: On the pronormality of subgroups of odd indices in the simple exceptional groups of Lie type, in preparation

  33. Kondratiev, A.S.: Normalizers of the Sylow 2-subgroups in finite simple groups. Math. Notes 78(3), 338–346 (2005)

    Article  MathSciNet  Google Scholar 

  34. Liebeck, M.W., Saxl, J.: The primitive permutation groups of odd degree. J. Lond. Math. Soc. II. Ser. 31(2), 250–264 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  35. Manzaeva, NCh.: Heritability of the property \({\mathscr {D}}_{\pi }\) by overgroups of \(\pi \)-Hall subgroups in the case where \(2\in \pi \). Algebra Logic 53(1), 17–28 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  36. Maslova, N.V.: Classification of maximal subgroups of odd index in finite simple classical groups. Proc. Steklov Inst. Math. 267(Suppl. 1), 164–183 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  37. Maslova, N.V.: Classification of maximal subgroups of odd index in finite simple classical groups: addendum. Siberian Electron. Math. Rep. 15, 707–718 (2018)

    MATH  Google Scholar 

  38. Maslova, N.V.: Classification of maximal subgroups of odd index in finite groups with alternating socle. Proc. Steklov Inst. Math. 285(Suppl. 1), 136–138 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  39. Mazurov, V.D., Khukhro, E.I. (eds.): The Kourovka Notebook. Unsolved Problems in Group Theory, 18th edn. Russian Academy of Sciences Siberian Division Institute of Mathematics, Novosibirsk (2014)

  40. Nesterov, M.N.: Pronormality of Hall subgroups in almost simple groups. Siberian Electron. Math. Rep. 12, 1032–1038 (2015)

    MathSciNet  MATH  Google Scholar 

  41. Nesterov, M.N.: On pronormality and strong pronormality of Hall subgroups. Siberian Math. J. 58(1), 128–133 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  42. Nesterov, M.N.: Arithmetic of conjugacy of \(p\)-complements. Algebra Logic 54(1), 36–47 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  43. Pálfy, P.: Isomorphism problem for relational structures with a cyclic automorphism. Acta Mathematica Academiae Seientiarum Hungaricae 34(3–4), 287–292 (1979)

    Article  Google Scholar 

  44. Pálfy, P.: Isomorphism problem for relational structures with a cyclic automorphism. Eur. J. Comb. 8, 35–43 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  45. Revin, D.O.: The \(D_\pi \)-property in finite simple groups. Algebra Logic 47(3), 210–227 (2008)

    Article  MathSciNet  Google Scholar 

  46. Revin, D.O.: The \(D_\pi \)-property in a class of finite groups. Algebra Logic 41(3), 187–206 (2002)

    Article  MathSciNet  Google Scholar 

  47. Revin, D.O., Vdovin, E.P.: Hall subgroups of finite groups. Contemp. Math. 402, 229–265 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  48. Revin, D.O., Vdovin, E.P.: On the number of classes of conjugate Hall subgroups in finite simple groups. J. Algebra 324(12), 3614–3652 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  49. Revin, D.O., Vdovin, E.P.: An existence criterion for Hall subgroups of finite groups. J. Group Theory 14(1), 93–101 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  50. Revin, D.O., Vdovin, E.P.: Frattini argument for Hall subgroups. J. Algebra 414, 95–104 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  51. Shemetkov, L.A.: Generalizations of Sylow’s theorem. Siberian Math. J. 44(6), 1127–1132 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  52. Skresanov, S.V.: The Wielandt–Hartley theorem for subnormal subgroups, in preparation

  53. Suzuki, M.: Group Theory II. Springer, New York (1986)

    Book  MATH  Google Scholar 

  54. Thompson, J.G.: Nonsolvable finite groups all of whose local subgroups are solvable. Bull. Am. Math. Soc. 74, 383–437 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  55. Tyshkevich, R.I.: Pronormal regular subgroups of the finite symmetric group. J. Sov. Math. 24(4), 470–475 (1984)

    Article  MATH  Google Scholar 

  56. Tyshkevich, R.I.: Relations admitting a transitive group of automorphisms. Math. USSR-Sbornik 26(2), 245–259 (1975)

    Article  MATH  Google Scholar 

  57. Vdovin, E.P.: Carter subgroups of finite almost simple groups. Algebra Logic 46(2), 90–119 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  58. Vdovin, E.P., Manzaeva, NCh., Revin, D.O.: On the heritability of the property \({\mathscr {D}}_{\pi }\) by subgroups. Proc. Steklov Inst. Math. 279(1), 130–138 (2012)

    Article  MATH  Google Scholar 

  59. Vdovin, E.P., Manzaeva, N.Ch., Revin, D.O.: The Hall property \({\mathscr {D}}_{\pi }\) is inherited by overgroups of \(\pi \)-Hall subgroups, to appear

  60. Vdovin, E.P., Nesterov, M.N., Revin, D.O.: On the pronormality of Hall subgroups in its normal closure. Algebra Logic 55(6), 451–457 (2017)

    Article  MATH  Google Scholar 

  61. Vdovin, E.P., Revin, D.O.: A conjugacy criterion for Hall subgroups in finite groups. Siberian Math. J. 51(3), 402–409 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  62. Vdovin, E.P., Revin, D.O.: Theorems of Sylow type. Russ. Math. Surv. 66(5), 829–870 (2011)

    Article  MATH  Google Scholar 

  63. Vdovin, E.P., Revin, D.O.: Abnormality criteria for \(p\)-complements. Algebra Logic 55(5), 347–353 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  64. Vdovin, E.P., Revin, D.O.: Pronormality of Hall subgroups in finite simple groups. Siberian Math. J. 53(3), 419–430 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  65. Vdovin, E.P., Revin, D.O.: On the pronormality of Hall subgroups. Siberian Math. J. 54(1), 22–28 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  66. Vdovin, E.P., Revin, D.O.: The existence of pronormal \(\pi \)-Hall subgroups in \(E_\pi \)-groups. Siberian Math. J. 56(3), 379–383 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  67. Wielandt, H.: Zum Satz von Sylow. II. Math. Z. 71(4), 461–462 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  68. Wielandt, H.: Entwicklungslinien in der Strukturtheorie der endlichen Gruppen. In: Proc. Intern. Congress Math., Edinburg, 1958. London: Cambridge Univ. Press, pp. 268–278. (1960)

  69. Wielandt, H.: Sur la Stucture des groupes composés, Séminare Dubriel-Pisot(Algèbre et Théorie des Nombres), 17e anée, 10 pp. 1963/64. N17

  70. Wielandt, H.: 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., pp. 161–173. (1980)

  71. Wielandt, H.: Zusammenghesetzte Gruppen endlicher Ordnung, Vorlesung an der Universität Tübingen im Wintersemester 1963/64. Helmut Wielandt: Mathematical Works, vol. 1, Group Theory (ed. B. Huppert and H. Schneider, de Gruyter, Berlin, 1994), pp. 607–516 (1994)

  72. Zenkov, V.I., Monakhov, V.S., Revin, D.O.: An analog for the Frattini factorization of finite groups. Algebra Logic 43(2), 102–108 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, People’s Republic of China

    Wenbin Guo & Danila O. Revin

  2. Sobolev Institute of Mathematics SB RAS, Novosibirsk, Russia

    Danila O. Revin

  3. Novosibirsk State University, Novosibirsk, Russia, 630090

    Danila O. Revin

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  1. Wenbin Guo
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  2. Danila O. Revin
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Correspondence to Wenbin Guo.

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The first author is supported by a NNSF Grant of China (Grant # 11771409) and Wu Wen-Tsun Key Laboratory of Mathematics of Chinese Academy of Sciences and by SB RAS Fundamental Research Program I.1.1 (project # 0314-2016-0001).

Appendix: Submaximal \(\pi \)-Subgroups of Minimal Simple Groups

Appendix: Submaximal \(\pi \)-Subgroups of Minimal Simple Groups

In Tables 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11 we use the following notation.

The conditions in the column ‘Cond.’ are necessary and sufficient for the existence and the \(\pi \)-submaximality of corresponding H. If a cell in this column is empty, then it means that the corresponding \(\pi \)-submaximal subgroup always exists.

In the column ‘H’ the structure of corresponding H is given.

The conditions in the column ‘is not \(\pi \)-max. if’ are necessary and sufficient for corresponding H to be not \(\pi \)-maximal in S. If either this column is skipped or a cell in this column is empty, then the corresponding subgroup is \(\pi \)-maximal.

A number n in the column ‘NCC’ is equal to the number of conjugacy classes of \(\pi \)-submaximal subgroups of S isomorphic to corresponding subgroup H, and if \(n>1\), then in the same column the action of \({{\mathrm{Aut}}}(S)\) on these classes is described.

The symbol ‘\(\checkmark \)’ in the column ‘Pro.’ means that the corresponding subgroup H is pronormal in S.

The symbol ‘\(\checkmark \)’ in the column ‘Intra.’ means that the corresponding subgroup H is intravariant in S. If a cell in this column is empty, then H is not intravariant.

1.1 The \(\pi \)-Submaximal Subgroups in \(S=L_2(q)\), Where \(q=2^p\), p is Prime, for \(\pi \) Such That \(|\pi \cap \pi (S)|>1\) and \(\pi (S){\varvec{\nsubseteq }}\pi \)

$$\begin{aligned} |S|=q(q-1)(q+1),\,\,\,\pi (S)=\{2\}\cup \pi (q-1)\cup \pi (q+1). \end{aligned}$$
Table 1 The \(\pi \)-submaximal subgroups of \(S=L_2(q)\), where \(q=2^p, p\) is a prime
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Table 2 The \(\pi \)-submaximal subgroups of \(S=L_2(q)\), where \(q=2^p\), p is a prime
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1.2 The \(\pi \)-Submaximal Subgroups in \(S=L_2(q)\), Where \(q=3^p\), p is Odd Prime, for \(\pi \) Such That \(|\pi \cap \pi (S)|>1\) and \(\pi (S){\varvec{\nsubseteq }}\pi \)

$$\begin{aligned} |S|=\frac{1}{2}q(q-1)(q+1),\,\,\,\pi (S)=\{3\}\cup \pi (q-1)\cup \pi (q+1). \end{aligned}$$
Table 3 The \(\pi \)-submaximal subgroups of \(S=L_2(q)\), where \(q=3^p,p\) is an odd prime
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Table 4 The \(\pi \)-submaximal subgroups of \(S=L_2(q)\), where \(q=3^p\), p is an odd prime
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1.3 The \(\pi \)-Submaximal Subgroups in \(S=L_2(q)\), Where q is a Prime, \(q^2\equiv -1\pmod 5\), for \(\pi \) Such That \(|\pi \cap \pi (S)|>1\) and \(\pi (S){\varvec{\nsubseteq }}\pi \)

$$\begin{aligned} |S|=\frac{1}{2}q(q-1)(q+1),\,\,\,\pi (S)=\{q\}\cup \pi (q-1)\cup \pi (q+1). \end{aligned}$$
Table 5 The \(\pi \)-submaximal subgroups of \(S=L_2(q)\), where \(q>3\) is a prime, \(q^2\equiv -1\pmod 5\)
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Table 6 The \(\pi \) -submaximal subgroups of \(S=L_2(q)\), where \(q>3\) is a prime, \(q^2\equiv -1\pmod 5\), in the case \(2\in \pi \)
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1.4 The Submaximal \(\pi \)-Subgroups in \(S=Sz(q)\), Where \(q=2^p\), p is Odd Prime, for \(\pi \) Such That \(|\pi \cap \pi (S)|>1\) and \(\pi (S){\varvec{\nsubseteq }}\pi \)

$$\begin{aligned} |S|= & {} q^2(q-1)(q^2+1)=q^2(q-1)(q-r+1)(q+r+1),\\ \text { where }r= & {} \sqrt{(2q)}=2^{(p+1)/2},\\ \pi (S)= & {} \{2\}\cup \pi (q-1)\cup \pi (q- r+1)\cup \pi (q+ r+1), \end{aligned}$$
Table 7 The \(\pi \)-submaximal subgroups of \(S=Sz(q)\), where \(q=2^p\), p is an odd prime
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Table 8 The \(\pi \)-submaximal subgroups of \(S=Sz(q)\), where \(q=2^p\), p is an odd prime
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1.5 The \(\pi \)-Submaximal Subgroups of \(S=L_3(3)\), for \(\pi \) Such That \(|\pi \cap \pi (S)|>1\) and \(\pi (S){\varvec{\nsubseteq }}\pi \)

$$\begin{aligned} |S|=2^4\cdot 3^3\cdot 13,\,\,\, \pi (S)=\{2,3,13\}, \end{aligned}$$
Table 9 The \(\pi \)-submaximal subgroups of \(S=L_3(3)\). Case: \(\pi \cap \pi (S)=\{3,13\}\)
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Table 10 The \(\pi \)-submaximal subgroups of \(S=L_3(3)\). Case: \(\pi \cap \pi (S)=\{2,13\}\)
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Table 11 The \(\pi \)-submaximal subgroups of \(S=L_3(3)\)
Full size table

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Guo, W., Revin, D.O. Pronormality and Submaximal \(\mathfrak {X}\)-Subgroups on Finite Groups. Commun. Math. Stat. 6, 289–317 (2018). https://doi.org/10.1007/s40304-018-0154-9

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