Advertisement

Springer Nature is making Coronavirus research free. View research | View latest news | Sign up for updates

Finite p-groups in which the number of subgroups of possible order is less than or equal to p 3

  • 27 Accesses

  • 6 Citations

Abstract

In this paper, groups of order p n in which the number of subgroups of possible order is less than or equal to p 3 are classified. It turns out that if p > 2, n ≥ 5, then the classification of groups of order p n in which the number of subgroups of possible order is less than or equal to p 3 and the classification of groups of order p n with a cyclic subgroup of index p 2 are the same.

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

References

  1. [1]

    Berkovich, Y., Groups of Prime Power Order I, Walter de Gruyter, Berlin, 2008.

  2. [2]

    Kulakoff, A., Über die Anzahl der eigentlichen Untergruppen und der Elemente von gegebener Ordnung in p-Gruppen, Math. Ann., 104, 1931, 778–793.

  3. [3]

    Tuan, H. F., An “Anzahl” theorem of Kulakoff’s type for p-groups, Sci. Rep. Nat. Tsing-Hua Univ. Ser. A., 5, 1948, 182–189.

  4. [4]

    Berkovich, Y., On the number of subgroups of given order in a finite p-group of exponent p, Proc. Amer. Math. Soc., 109, 1990, 875–879.

  5. [5]

    Xu, M. Y., Some Problems on Finite p-Groups (in Chinese), Adv. Math., 14(3), 1985, 205–226.

  6. [6]

    Chen, Y. H. and Cao, H. P., The complete classification of p-group with p + 1 nontrivial subgroups of each order (in Chinese), J. Southwest Univ., 29(2), 2007, 11–14.

  7. [7]

    Zhang, Q. H. and Qu, H. P., On Hua-Tuan’s conjecture, Sci. in China Ser. A, Math., 52(2), 2009, 389–393.

  8. [8]

    Huppert, B., Endliche Gruppen I, Springer-Verlag, Berlin, 1967.

  9. [9]

    Taussky, O., Remark on the class field tower, J. London Math. Soc., 12, 1937, 82–85.

  10. [10]

    Rédei, L., Das schiefe product in der Gruppentheorie, Comm. Math. Helvet, 20, 1947, 225–267.

  11. [11]

    Hua, L. K. and 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. Ser. A., 4, 1940, 145–151.

  12. [12]

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

  13. [13]

    Li, L. L., Qu, H. P. and Chen, G. Y., Central extension of inner abelian p-groups I (in Chinese), Acta Math. Sinica, 53(4), 2010, to appear.

Download references

Author information

Correspondence to Qinhai Zhang.

Additional information

Project supported by the National Natural Science Foundation of China (No. 10671114), the Shanxi Provincial Natural Science Foundation of China (No. 2008012001) and the Returned Abroad-Student Fund of Shanxi Province (No. [2007]13-56).

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Qu, H., Sun, Y. & Zhang, Q. Finite p-groups in which the number of subgroups of possible order is less than or equal to p 3 . Chin. Ann. Math. Ser. B 31, 497–506 (2010). https://doi.org/10.1007/s11401-010-0590-7

Download citation

Keywords

  • Inner abelian p-groups
  • Metacyclic p-groups
  • Groups of order p n with a cyclic subgroup of index p 2
  • The number of subgroups

2000 MR Subject Classification

  • 20D15