Ukrainian Mathematical Journal

, Volume 63, Issue 11, pp 1745–1755 | Cite as

Q-Permutable subgroups of finite groups

  • Z. Pu
  • L. MiaoEmail author

A subgroup H of a group G is called Q-permutable in G if there exists a subgroup B of G such that (1) G = HB and (2) if H 1 is a maximal subgroup of H containing H QG , then H 1 B = BH 1 < G, where H QG is the largest permutable subgroup of G contained in H. In this paper, we prove the following statement: Let \( \mathcal{F} \) be a saturated formation containing \( \mathcal{U} \) and let G be a group with a normal subgroup H such that \( {{G} \left/ {H} \right.} \in \mathcal{F} \). If every maximal subgroup of every noncyclic Sylow subgroup of F*(H) having no supersolvable supplement in G is Q-permutable in G, then \( G \in \mathcal{F} \).


Normal Subgroup Maximal Subgroup Sylow Subgroup Minimal Normal Subgroup Saturated Formation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. Ballester-Bolinches, Y. Wang, and X. Guo, “C-supplemented subgroups of finite groups,” Glasgow Math. J., 42, 383–389 (2000).MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    K. Doerk and T. Hawkes, Finite Soluble Groups, de Gruyter, Berlin–New York (1992).zbMATHCrossRefGoogle Scholar
  3. 3.
    F. Gross, “Conjugacy of odd order Hall subgroups,” Bull. London Math. Soc., 19, 311–319 (1987).MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    W. Guo, The Theory of Classes of Groups, Science Press–Kluwer, Beijing (2000).zbMATHGoogle Scholar
  5. 5.
    P. Hall, “A characteristic property of soluble groups,” J. London Math. Soc., 12, 188–200 (1937).Google Scholar
  6. 6.
    B. Huppert, Endliche Gruppen I, Springer, Berlin (1967).zbMATHCrossRefGoogle Scholar
  7. 7.
    B. Huppert and N. Blackburn, Finite Groups III, Springer, Berlin (1982).zbMATHCrossRefGoogle Scholar
  8. 8.
    O. H. Kegel, “On Huppert’s characterization of finite supersoluble groups,” in: Proceedings of the International Conference on the Theory of Groups (Canberra, 1965), New York (1967), pp. 209–215.Google Scholar
  9. 9.
    O. H. Kegel, “Produkte nilpotenter gruppen,” Arch. Math. (Basel), 12, 90–93 (1961).MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    L. Miao and W. Lempken, “On \( \mathcal{M} \)-supplemented subgroups of finite groups,” J. Group Theory, 12, No. 2, 271–287 (2009).MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    K. Nakamura, “Beziehungen zwischen den Strukturen von Normalteiler und Quasinormalteiler,” Osaka J. Math., 7, 321–322 (1970).MathSciNetzbMATHGoogle Scholar
  12. 12.
    D. J. Robinson, A Course in the Theory of Groups, Springer, Berlin (1993).Google Scholar
  13. 13.
    A. N. Skiba, “On weakly s-permutable subgroups of finite groups,” J. Algebra, 315, 192–209 (2007).MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Y. Wang, “C-normality of groups and its properties,” J. Algebra, 78, 101–108 (1996).Google Scholar
  15. 15.
    Y. Wang, H. Wei, and Y. Li, “A generalization of Kramer’s theorem and its applications,” Bull. Austral. Math. Soc., 65, 467–475 (2002).MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    M. Xu, An Introduction to Finite Groups, Science Press, Beijing (1999) [in Chinese].Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2012

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsHexi UniversityGansuChina
  2. 2.School of Mathematical SciencesYangzhou UniversityYangzhouChina

Personalised recommendations