Skip to main content
SpringerLink
Log in

Structure of k-Closures of Finite Nilpotent Permutation Groups

  • Published:
Algebra and Logic Aims and scope

Let G be a permutation group of a set Ω and k be a positive integer. The k-closure of G is the greatest (w.r.t. inclusion) subgroup G(k) in Sym(Ω) which has the same orbits as has G under the componentwise action on the set Ωk. It is proved that the k-closure of a finite nilpotent group coincides with the direct product of k-closures of all of its Sylow subgroups.

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. H. Wielandt, “Permutation groups through invariant relation and invariant functions,” Lect. Notes Dept. Math. Ohio St. Univ., Columbus, 1969, in H. Wielandt, Mathematische Werke. Mathematical works. Vol. 1: Group Theory, B. Huppert and H. Schneider (eds.), Walter de Gruyter, Berlin (1994), pp. 237-266.

  2. D. Churikov and I. Ponomarenko, “On 2-closed abelian permutation groups,” submitted to Comm. Algebra (2020), see also arXiv:2011.12011 [math.GR].

  3. D. Churikov and Ch. E. Praeger, “Finite totally k-closed groups,” Tr. Inst. Mat. Mech. UrO RAN, 27, No. 1, 240-245 (2021).

    MathSciNet  Google Scholar 

  4. H. Wielandt, Finite Permutation Groups, Academic Press, New York (1964).

    MATH  Google Scholar 

  5. E. A. O’Brien, I. Ponomarenko, A. V. Vasil’ev, and E. Vdovin, “The 3-closure of a solvable permutation group is solvable,” arXiv:2012.14166 [math.GR].

  6. S. Evdokimov and I. Ponomarenko, “Two-closure of odd permutation group in polynomial time,” Discr. Math., 235, Nos. 1-3, 221-232 (2001).

    Article  MathSciNet  Google Scholar 

  7. P. J. Cameron, M. Giudici, G. A. Jones, W. M. Kantor, M. H. Klin, D. Marušič, and L. A. Nowitz, “Transitive permutation groups without semiregular subgroups,” J. London Math. Soc., II. Ser., 66, No. 2, 325-333 (2002).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    D. V. Churikov

  2. Novosibirsk State University, Novosibirsk, Russia

    D. V. Churikov

Authors
  1. D. V. Churikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. V. Churikov.

Additional information

Translated from Algebra i Logika, Vol. 60, No. 2, pp. 231-239, March-April, 2021. Russian https://doi.org/10.33048/alglog.2021.60.208.

Supported by Mathematical Center in Akademgorodok, Agreement with RF Ministry of Education and Science No. 075-15-2019-1613.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Churikov, D.V. Structure of k-Closures of Finite Nilpotent Permutation Groups. Algebra Logic 60, 154–159 (2021). https://doi.org/10.1007/s10469-021-09637-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10469-021-09637-9

Keywords

Search

Navigation