Skip to main content
SpringerLink
Log in

The intersection of subgroups of finite p-groups

  • Published:
Archiv der Mathematik Aims and scope Submit manuscript

Abstract

For a finite p-group G and a positive integer k let I k (G) denote the intersection of all subgroups of G of order p k. This paper classifies the finite p-groups G with \({{I}_k(G)\cong C_{p^{k-1}}}\) for primes p > 2. We also show that for any k, α ≥ 0 with 2(α + 1) ≤ k ≤ n−α the groups G of order p n with \({{I}_k(G)\cong C_{p^{k-\alpha}}}\) are exactly the groups of exponent p n-α.

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. Bai S.W.: A classification of 2-groups with a cyclic subgroup of index 22. J. Heilongjiang University (Natural Sciences) 2, 74–85 (1985) 69 (in Chinese)

    Google Scholar 

  2. Berkovich Y.: Groups of Prime Power Order I. Walter de Gruyter, Berlin New York (2008)

    Google Scholar 

  3. Blackburn N.: Generalizations of certain elementary theorems on p-groups. Proc. London Math. Soc. (3) 11, 1–22 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bosma W., Cannon J., Playoust C.: The Magma algebra system I: The user language. J. Symbolic Comput. 24, 235–265 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  5. W. Burnside, Theory of Groups of Finite Order, Cambridge University Press, 1897.

  6. Hua L.-K.: Some “Anzahl” theorems for groups of prime power orders. Sci. Rep. Nat. Tsing Hua Univ. 4, 313–327 (1947)

    MathSciNet  Google Scholar 

  7. Hua L.K., Tuan H. F.: Determination of the groups of odd-prime-power order p n which contain a cyclic subgroup of index p 2. Sci. Rep. Nat. Tsing. Hua Univ. A 4, 145–154 (1940)

    MathSciNet  Google Scholar 

  8. Huppert B.: Endliche Gruppen I. Springer-Verlag, Berlin Heidelberg New York (1967)

    MATH  Google Scholar 

  9. C. R. Leedham-Green and S. McKay, The structure of groups of prime power order. LMS Monographs new series 27, Oxford University Press 2002.

  10. M. Y. Xu and H. P. Qu, Finite p-groups, Beijing University Press, 2010. (in Chinese)

  11. Xu M.Y.: A theorem on metabelian p-groups and some consequences. Chin. Ann. Math. 5, 1–6 (1984)

    MATH  Google Scholar 

  12. Q. H. Zhang and P. J. Li, Finite p-groups with a cyclic subgroup of index p 3, submitted.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Shanxi Normal University, Linfen, 041004, Shanxi, People’s Republic of China

    Qinhai Zhang & Junjun Wei

Authors
  1. Qinhai Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Junjun Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qinhai Zhang.

Additional information

This work was supported by NSFC (no. 11071150), by NSF of Shanxi Province (no. 2008012001) and The Returned Abroad-student Found of Shanxi Province (no. [2007] 13-56).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, Q., Wei, J. The intersection of subgroups of finite p-groups. Arch. Math. 96, 9–17 (2011). https://doi.org/10.1007/s00013-010-0194-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00013-010-0194-1

Mathematics Subject Classification (2000)

Keywords

Search

Navigation