Advertisement

Indices of Maximal Invariant Subgroups and Solvability of Finite Groups

  • Changguo ShaoEmail author
  • Antonio Beltrán
Article
  • 71 Downloads

Abstract

Let A and G be finite groups and suppose that A acts coprimely on G via automorphisms. We study the solvability and supersolvability of G when certain proper maximal A-invariant subgroups of G have prime index or when they have certain prime power indices in G.

Keywords

Finite groups maximal subgroups subgroup index coprime action group action on groups 

Mathematics Subject Classification

20D20 20D15 

Notes

Acknowledgements

Part of this work was written while the first author was doing a research stay during 2017 at the Universidad Jaume I of Castellón, Spain. He would like to thank the Mathematics Department of the institution and A. Beltrán for their hospitality. The first author is supported by the NNSF of China (No. 11301218), the Nature Science Fund of Shandong Province (No. ZR2018AM020) and the Opening Project of Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing (No. 2018QZJ04). The second author was partially supported by the Valencian Government, Proyecto PROMETEOII/2015/011 and also by Universitat Jaume I, Grant P11B-2015-77.

References

  1. 1.
    Lu, L., Pang, L., Zhong, X.: Finite groups with non-nilpotent maximal subgroups. Monatsh. Math. 171, 425–431 (2013)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Li, X.H.: A characterization of the finite simple groups with set of indices of their maximal subgroups. Sci. China Math. 47(4), 508522 (2004)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Huppert, B.: Endliche Gruppen. I. Springer, Berlin (1967)CrossRefGoogle Scholar
  4. 4.
    Guralnick, R.M.: Subgroups of prime power index in a simple group. J. Algebra 81(2), 304–311 (1983)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Beltrán, A., Shao, C.: On the number of invariant Sylow subgroups under coprime action. J. Algebra 490, 380–389 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Beltrán, A., Shao, C.: Restrictions on maximal invariant subgroups implying solvability of finite groups. Annali di Matematica Pura ed Applicata (2018).  https://doi.org/10.1007/s10231-018-0777-1 CrossRefzbMATHGoogle Scholar
  7. 7.
    Kurzweil, H., Stellmacher, B.: The Theory of Finite Groups. Springer, Berlin (2004). An introductionCrossRefGoogle Scholar
  8. 8.
    Monakhov, V.S., Tyutyanov, V.N.: On finite groups with some subgroups of prime indices. Sib. Math. J. 48(4), 666–668 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Huppert, B., Blackburn, N.: Finite Groups. III. Springer, Berlin (1982)CrossRefGoogle Scholar
  10. 10.
    Zsigmondy, K.: Zur Theorie der Potenzreste. Monatsh. Math. Phys. 3, 265–284 (1892)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: Atlas of Finite Groups. Oxford University Press, London (1985)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of JinanShandongChina
  2. 2.Departamento de MatemáticasUniversidad Jaume ICastellónSpain

Personalised recommendations