Skip to main content
SpringerLink
Log in

Shunkov Groups Saturated with Almost Simple Groups

  • Published:
Algebra and Logic Aims and scope

A group G is called a Shunkov group (a conjugate biprimitive finite group) if, for any of its finite subgroups H in the factor group NG(H)/H, every two conjugate elements of prime order generate a finite subgroup. We say that a group is saturated with groups from the set 𝔐 if any finite subgroup of the given group is contained in its subgroup isomorphic to some group in 𝔐. We show that a Shunkov group G which is saturated with groups from the set 𝔐 possessing specific properties, and contains an involution z with the property that the centralizer CG(z) has only finitely many elements of finite order will have a periodic part isomorphic to one of the groups in 𝔐. In particular, a Shunkov group G that is saturated with finite almost simple groups and contains an involution z with the property that the centralizer CG(z) has only finitely many elements of finite order will have a periodic part isomorphic to a finite almost simple group.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. W. Guo, D. V. Lytkina, and V. D. Mazurov, “Periodic groups saturated with finite simple groups L4(q),” Algebra and Logic, 60, No. 6, 360-365 (2021).

    Article  MathSciNet  Google Scholar 

  2. D. V. Lytkina, A. I. Sozutov, and A. A. Shlepkin, “2-rank two periodic groups saturated with finite simple groups,” Sib. El. Mat. Izv., 15, 786-796 (2018)

    Google Scholar 

  3. D. V. Lytkina and V. D. Mazurov, “Locally finite periodic groups saturated with finite simple orthogonal groups of odd dimension,” Sib. Math. J., 62, No. 3, 462-467 (2021).

    Article  MathSciNet  Google Scholar 

  4. B. E. Durakov and A. I. Sozutov, “On periodic groups saturated with finite Frobenius groups,” Izv. Irkutsk Gos. Univ., Mat., 35, 73-86 (2021).

  5. A. A. Shlepkin, “Groups with a strongly embedded subgroup saturated with finite simple non-abelian groups,” Izv. Irkutsk Gos. Univ., Mat., 31, 132-141 (2020).

  6. V. P. Shunkov, “On periodic groups with an almost regular involution,” Algebra and Logic, 11, No. 4, 260-272 (1972).

    Article  MathSciNet  Google Scholar 

  7. V. V. Belyaev, “Groups with an almost regular involution,” Algebra and Logic, 26, No. 5, 315-317 (1987).

    Article  MathSciNet  Google Scholar 

  8. A. I. Sozutov, “On groups with almost perfect involution,” Trudy Inst. Mat. Mekh. UrO RAN, 13, No. 1, 183-190 (2007).

    MathSciNet  Google Scholar 

  9. A. A. Shlepkin, “On a sufficient condition when an infinite group is not simple,” Zh. SFU, Ser. Mat. Fiz., 11, No. 1, 103-107 (2018).

  10. A. K. Shlyopkin, “Some periodic groups saturated with finite simple subgroups,” Mat. Trudy, 1, No. 1, 129-138 (1998).

    MathSciNet  Google Scholar 

  11. M. Aschbacher, “On the maximal subgroups of the finite classical groups,” Invent. Math., 76, No. 3, 469-514 (1984).

    Article  MathSciNet  Google Scholar 

  12. A. K. Shlyopkin, “Conjugate biprimitively finite groups with the primary minimal condition,” Algebra and Logic, 22, No. 2, 165-169 (1983).

    Article  Google Scholar 

  13. P. J. Cameron and N. V. Maslova, “Criterion of unrecognizability of a finite group by its Gruenberg–Kegel graph,” J. Alg., 607, Part A, 186-213 (2022).

  14. P. Fong, “On orders of finite groups and centralizers of p-elements,” Osaka J. Math., 13, 483-489 (1976).

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Krasovskii Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, Yekaterinburg, Russia

    N. V. Maslova

  2. El’tsyn Ural Federal University, Yekaterinburg, Russia

    N. V. Maslova

  3. Siberian Federal University, Krasnoyarsk, Russia

    A. A. Shlepkin

Authors
  1. N. V. Maslova
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. A. Shlepkin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. V. Maslova.

Additional information

Dedicated to V. D. Mazurov on the occasion of his 80th birthday

Supported by RFBR, project No. 20-01-00456 (Thm. 2). (N. V. Maslova)

Supported by Russian Science Foundation, project No. 19-71-10017-P (Thm. 1). (A. A. Shlepkin)

Translated from Algebra i Logika, Vol. 62, No. 1, pp. 93-101, January-February, 2023. Russian https://doi.org/10.33048/alglog.2023.62.106.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Maslova, N.V., Shlepkin, A.A. Shunkov Groups Saturated with Almost Simple Groups. Algebra Logic 62, 66–71 (2023). https://doi.org/10.1007/s10469-023-09725-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10469-023-09725-y

Search

Navigation