Archiv der Mathematik

, Volume 110, Issue 5, pp 427–432 | Cite as

Carter subgroups and Fitting heights of finite groups

  • Wenbin Guo
  • E. P. Vdovin


Let G be a finite group possessing a Carter subgroup K. Denote by \(\mathbf {h}(G)\) the Fitting height of G, by \(\mathbf {h}^*(G)\) the generalized Fitting height of G, and by \(\ell (K)\) the number of composition factors of K, that is, the number of prime divisors of the order of K with multiplicities. In 1969, E. C. Dade proved that if G is solvable, then \(\mathbf {h}(G)\) is bounded in terms of \(\ell (K)\). In this paper, we show that \(\mathbf {h}^*(G)\) is bounded in terms of \(\ell (K)\) as well.


Finite group Carter subgroup Generalized Fitting subgroup Generalized Fitting height 

Mathematics Subject Classification

20D25 20D30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors express gratitude to Professor E.I.Khukhro for his comments and suggestions.


  1. 1.
    M. Aschbacher, Finite Group Theory, Cambridge Univ. Press, Cambridge, 1986.zbMATHGoogle Scholar
  2. 2.
    E.C. Dade, Carter subgroups and Fitting heights of finite solvable groups, Illinois J. Math. 13 (1969), 449–514.MathSciNetzbMATHGoogle Scholar
  3. 3.
    T.O. Hawkes, Two applications of twisted wreath products to finite soluble groups, Trans. Amer. Math. Soc. 214 (1975), 325–335.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    B. Huppert and N. Blackburn, Finite Groups III, Springer-Verlag, Berlin-New York, 1982.CrossRefzbMATHGoogle Scholar
  5. 5.
    E.I. Khukhro and P. Shumyatsky, On the length of finite groups and of fixed points, Proc. Am. Math. Soc. 143 (2015), 3781–3790.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    J. G. Thompson, Automorphisms of solvable groups, J. Algebra 1 (1964), 259–267.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    A. Turull, Character theory and length problems, In: Finite and Locally Finite Groups (Istanbul, 1994), 377–400, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 471, Kluwer Acad. Publ., Dordrecht, 1995.Google Scholar
  8. 8.
    E.P. Vdovin, Carter subgroups of almost simple groups, Algebra and Logic 46 (2007), 90–119.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    E.P. Vdovin, On the existence of Carter subgroups, Proc. Stekl. Inst. Math. 257 (2008), 195–204.MathSciNetCrossRefGoogle Scholar
  10. 10.
    E.P. Vdovin, On the conjugacy problem for Carter subgroups, Sib. Math. Journal 47 (2006), 597-600.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Science and Technology of ChinaHefeiPeople’s Republic of China
  2. 2.Sobolev Institute of Mathematics and Novosibirsk State UniversityNovosibirskRussia

Personalised recommendations