On σ-Subnormal Subgroups of Finite Groups

Abstract

Let p be a prime and let σ = {{p}, {p}′} be a partition of the set ℙ of all primes. We prove the following conjecture by Skiba: If each complete Hall set of type σ in a finite group G is reducible to some subgroup H of G then H is σ-subnormal in G.

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References

  1. 1.

    Kegel O. H., “Sylow-Gruppen und Subnormalteiler endlicher Gruppen,” Math. Z., Bd 78, 205–221 (1962).

    MathSciNet  Article  Google Scholar 

  2. 2.

    Wielandt H., “Zusammengesetzte Gruppen: Hölders Programm heute,” Proc. Pure Math., vol. 37, 161–173 (1980).

    Article  Google Scholar 

  3. 3.

    Kleidman P. B., “A proof of the Kegel-Wielandt conjecture on subnormal subgroups,” Ann. Math., vol. 133, no. 2, 369–428 (1991).

    MathSciNet  Article  Google Scholar 

  4. 4.

    Mazurov V. D. and Khukhro E. I. (eds.), The Kourovka Notebook: Unsolved Problems in Group Theory, Sobolev Inst. Math., Novosibirsk (2018).

    Google Scholar 

  5. 5.

    Skiba A. N., “On some results in the theory of finite partially soluble groups,” Commun. Math. Stat., vol. 4, no. 3, 281–309 (2016).

    MathSciNet  Article  Google Scholar 

  6. 6.

    Wielandt H., “Eine Verallgemeinerung der invarianten Untergruppen,” Math. Z., Bd 45, 209–244 (1939).

    MathSciNet  Article  Google Scholar 

  7. 7.

    Skiba A. N., “On σ-properties of finite groups. I,” Probl. Fiz. Math. Tekh., no. 4, 89–96 (2014).

    MATH  Google Scholar 

  8. 8.

    Skiba A. N., “On σ-subnormal and σ-permutable subgroups of finite groups,” J. Algebra, vol. 436, 1–16 (2015).

    MathSciNet  Article  Google Scholar 

  9. 9.

    Kazarin L. S., “On a product of finite groups,” Dokl. Akad. Nauk SSSR, vol. 269, no. 3, 528–531 (1983).

    MathSciNet  Google Scholar 

  10. 10.

    Doerk K. and Hawkes T., Finite Soluble Groups, Walter de Gruyter, Berlin and New York (1992).

    Google Scholar 

  11. 11.

    Revin D. O. and Vdovin E. P., “Hall subgroups of finite groups,” in: Ischia Group Theory 2004: Proc. Conf. in Honor of Marcel Herzog (Naples, Italy, 2004), Amer. Math. Soc., Providence, 2006, 229–265.

    Google Scholar 

  12. 12.

    Guralnick R., Kleidman P. B., and Lyons R., “Sylow p-subgroups and subnormal subgroups of finite groups,” Proc. London Math. Soc., vol. 66, no. 3, 129–151 (1993).

    MathSciNet  Article  Google Scholar 

  13. 13.

    Zsigmondy K., “Zur Theorie der Potenzreste,” Monath. Math. Phys., vol. 3, 265–284 (1892).

    MathSciNet  Article  Google Scholar 

Download references

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Correspondence to S. F. Kamornikov or V. N. Tyutyanov.

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Russian Text © The Author(s), 2020, published in Sibirskii Matematicheskii Zhurnal, 2020, Vol. 61, No. 2, pp. 337–343.

S. F. Kamornikov was supported by the Ministry of Education of the Republic of Belarus (Grant GR 20191056).

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Kamornikov, S.F., Tyutyanov, V.N. On σ-Subnormal Subgroups of Finite Groups. Sib Math J 61, 266–270 (2020). https://doi.org/10.1134/S0037446620020093

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Keywords

  • finite group
  • σ-subnormal subgroup
  • Hall subgroup
  • complete Hall set