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Injectors in fitting sets of finite groups

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Abstract

A set of subgroups ℱ of a finite group G is referred to as a Fitting set if it is closed with respect to taking normal subgroups, products of normal ℱ-subgroups, and inner automorphisms of G. A Fitting set ℱ of a group G is said to be π-saturated if H ∈ ℱ for every subgroup H in G such that O π’ (H) ∈ ℱ. In the paper, it is proved that, if ℱ is a π-saturated Fitting set of a π-solvable group G, then there are ℱ-injectors in G and every two of them are conjugate.

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References

  1. B. Fischer, W. Gaschütz, and B. Hartley, “Injektoren endlicher auflösbarer Gruppen,” Math. Z. 102 (5), 337–339 (1967).

    Article  MATH  MathSciNet  Google Scholar 

  2. L. A. Shemetkov, “Subgroups of π-solvable groups,” in Finite Groups (Nauka i Tekhnika, Minsk, 1975), pp. 207–212 [in Russian].

    Google Scholar 

  3. W. Anderson, “Injectors in finite solvable groups,” J. Algebra 36 (3), 333–338 (1975).

    Article  MATH  MathSciNet  Google Scholar 

  4. K. Doerk and T. Hawkes, Finite Soluble Groups, in de Gruyter Exp. Math. (Walter de Gruyter, Berlin, 1992), Vol. 4 [in Russian].

  5. L. A. Shemetkov, Formations of Finite Groups, in Modern Algebra (Nauka, Moscow, 1978) [in Russian].

    MATH  Google Scholar 

  6. A. Ballester-Bolinches and L. M. Ezquerro, Classes of Finite Groups, in Math. Appl. (Springer) (Springer, Dordrecht, 2006), Vol. 584 [in Russian].

  7. I. Hawthorn, “The existence and uniqueness of injectors for Fitting sets of solvable groups,” Proc. Amer. Math. Soc. 126 (8), 2229–2230 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  8. P. A. Gol’berg, “Hall θ-bases of finite groups,” Izv. Vyss. Uchebn. Zaved.Matematika, No. 1, 36–43 (1961).

    MathSciNet  Google Scholar 

  9. S. N. Vorob’ev and E. N. Zalesskaya, “An analog of Shemetkov’s conjecture for Fischer classes of finite groups,” Sib. Math. J. 54 (5), 989–999 (2013) [Sib.Math. J. 54 (5), 790–797 (2013)].

    MathSciNet  Google Scholar 

  10. W. Guo, “Injectors of finite groups,” Chinese Ann. Math. Ser. A 18 (2), 145–148 (1997).

    MATH  MathSciNet  Google Scholar 

  11. V. G. Sementovskii, “Injectors of finite groups,” in Investigation of the Normal and Subgroup Structure of Finite Groups (“Nauka i Tekhnika”, Minsk, 1984), pp. 166–170 [in Russian].

    Google Scholar 

  12. W. Guo and N. T. Vorob’ev, “On injectors of finite soluble groups,” Comm. Algebra 36 (9), 3200–3208 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  13. Y. Liu, W. Guo, and N. T. Vorob’ev,, “Description of ℱ-injectors of Finite Soluble Groups,” Math. Sci. Res. J. 12 (1), 17–22 (2008).

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Masherov Vitebsk State University, Vitebsk, Belarus

    N. T. Vorob’ev & M. G. Semenov

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  1. N. T. Vorob’ev
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  2. M. G. Semenov
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Correspondence to N. T. Vorob’ev.

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Vorob’ev, N.T., Semenov, M.G. Injectors in fitting sets of finite groups. Math Notes 97, 521–530 (2015). https://doi.org/10.1134/S0001434615030244

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  • DOI: https://doi.org/10.1134/S0001434615030244

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