Advertisement

Monatshefte für Mathematik

, Volume 171, Issue 2, pp 205–216 | Cite as

Finite groups with some NR-subgroups or \({\mathcal{H}}\)-subgroups

  • Izabela Agata MalinowskaEmail author
Open Access
Article
  • 447 Downloads

Abstract

Berkovich investigated the following concept: a subgroup H of a finite group G is called an NR-subgroup (Normal Restriction) if whenever \({K \trianglelefteq H}\), then \({K^G \cap H = K}\), where K G is the normal closure of K in G. Bianchi, Gillio Berta Mauri, Herzog and Verardi proved a characterization of soluble T-groups by means of \({\mathcal{H}}\)-subgroups: a subgroup H of G is said to be an \({\mathcal{H}}\)-subgroup of G if \({H^g \cap N_G(H) \leq H}\) for all \({g \in G}\). In this article we give new characterizations of finite soluble PST-groups in terms of NR-subgroups or \({\mathcal{H}}\)-subgroups. We will show that they are different from the ones given by Ballester-Bolinches, Esteban-Romero and Pedraza-Aguilera. Robinson established the structure of minimal non-PST-groups. We give the classification of groups all of whose second maximal subgroups (of even order) are soluble PST-groups.

Keywords

NR-subgroups \({\mathcal{H}}\)-subgroups T-groups PT-groups PST-groups 

Mathematics Subject Classification (2000)

20D10 20D20 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

References

  1. 1.
    Asaad M., Heliel A.A.: Finite groups in which normality is a transitive relation. Arch. Math. (Basel) 76(5), 321–325 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Asaad M.: On p-nilpotence and supersolvability of finite groups. Comm. Algebra 34, 4217–4224 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Ballester-Bolinches A., Esteban-Romero R.: Sylow permutable subnormal subgroups of finite groups. II. Bull Austral. Math. Soc. 64(3), 479–486 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Ballester-Bolinches A., Esteban-Romero R., Asaad M.: Products of finite groups. Walter de Gruyter GmbH & Co, KG, Berlin (2010)zbMATHCrossRefGoogle Scholar
  5. 5.
    Ballester-Bolinches A., Esteban-Romero R., Li Y.: On self-normalizing subgroups of finite groups. J. Group Theory 13(1), 143–149 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Ballester-Bolinches A., Esteban-Romero R., Pedraza-Aguilera M.C.: On a class of p-soluble groups. Algebra Colloq. 12(2), 263–267 (2005)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Berkovich Y.: Subgroups with the character restriction property and related topics. Houston J. Math. 24, 631–638 (1998)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bianchi M., Gillio Berta Mauri A., Herzog M., Verardi L.: On finite solvable groups in which normality is a transitive relation. J. Group Theory 3, 147–156 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Goldschmidt D.: Strongly closed 2-subgroups of finite groups. Ann. Math. 102(2), 475–489 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Huppert B.: Endliche Gruppen I. Springer, Berlin (1967)zbMATHCrossRefGoogle Scholar
  11. 11.
    Huppert B.: Normalteiler und maximal Untergruppen endlicher gruppen. Math. Z. 60, 409–434 (1954)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Isaacs I.M.: Subgroups with the character restriction property. J. Algebra 100, 403–420 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Li S., Zhao Y.: Some finite nonsolvable groups characterized by their solvable subgroups. Acta Math. Sinica (N.S.) 4, 5–13 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Ramadan M.: Finite groups in which permutability is a transitive relation on their Frattini factor groups. Acta Math. Hungar 114(3), 187–193 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Robinson D.J.S.: A course in the theory of groups. Springer, New York (1996)CrossRefGoogle Scholar
  16. 16.
    Robinson D.J.S.: Groups which are minimal with respect to normality being transitive. Pac. J. Math. 31, 777–789 (1969)zbMATHCrossRefGoogle Scholar
  17. 17.
    Robinson D.J.S.: Minimality and Sylow-permutability in locally finite groups. Ukrainian Math. J. 54(6), 1038–1049 (2002)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Suzuki M.: Group theory I. Springer, Berlin (1982)zbMATHGoogle Scholar
  19. 19.
    Tong-Viet H.P.: Groups with normal restriction property. Arch. Math. (Basel) 93, 199–203 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    van der Waall R.W., Fransman A.: On products of groups for which normality is a transitive relation on their Frattini factor groups. Quaestiones Math. 19(1–2), 59–82 (1996)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2012

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of BiałystokBiałystokPoland

Personalised recommendations