Finite p-groups whose non-normal subgroups have few orders

Research Article


Suppose that G is a finite p-group. If G is not a Dedekind group, then G has a non-normal subgroup. We use pM(G) and pm(G) to denote the maximum and minimum of the orders of the non-normal subgroups of G, respectively. In this paper, we classify groups G such that M(G) < 2m(G) − 1. As a by-product, we also classify p-groups whose orders of non-normal subgroups are p k and pk+1.


Finite p-groups meta-hamiltonian p-groups non-normal subgroups 




Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11471198, 11771258).


  1. 1.
    An L, Li L, Qu H, Zhang Q. Finite p-groups with a minimal non-abelian subgroup of index p (II). Sci China Math, 2014, 57: 737–753MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    An L, Zhang Q. Finite metahamiltonian p-groups. J Algebra, 2015, 442: 23–35MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Berkovich Y. Groups of Prime Power Order, Vol. 1. Berlin: Walter de Gruyter, 2008MATHGoogle Scholar
  4. 4.
    Fang X, An L. The classification of finite metahamiltonian p-groups. 1310.5509v2Google Scholar
  5. 5.
    Passman D S. Nonnormal subgroups of p-groups. J Algebra, 1970, 15: 352–370MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Rédei L. Das “schiefe Produkt” in der Gruppentheorie mit Anwendung auf die endlichen nichtkommutativen Gruppen mit lauter kommutativen echten Untergruppen und die Ordnungszahlen, zu denen nur kommutative Gruppen gehören (German). Comment Math Helvet, 1947, 20: 225–264CrossRefMATHGoogle Scholar
  7. 7.
    Xu M, An L, Zhang Q. Finite p-groups all of whose non-abelian proper subgroups are generated by two elements. J Algebra, 2008, 319: 3603–3620MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Zhang Q, Guo X, Qu H, Xu M. Finite group which have many normal subgroups. J Korean Math Soc, 2009, 46(6): 1165–1178MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Zhang Q, Li X, Su M. Finite p-groups whose nonnormal subgroups have orders at most p 3. Front Math China, 2014, 9(5): 1169–1194MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Zhang Q, Su M. Finite 2-groups whose nonnormal subgroups have orders at most 23. Front Math China, 2012, 7(5): 971–1003MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Zhang Q, Zhao L, Li M, Shen Y. Finite p-groups all of whose subgroups of index p 3 are abelian. Commun Math Stat, 2015, 3: 69–162MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsShanxi Normal UniversityLinfenChina

Personalised recommendations