On Centralizers of Locally Finite Simple Groups

  • Mattia Brescia
  • Alessio RussoEmail author


The aim of this article is to prove the following theorem. Let G be any infinite simple locally finite group. Then, either G is isomorphic to \(\mathrm{{PSL}}(2,F)\), where F is an infinite locally finite field, or G contains a subgroup which is the direct product of an infinite abelian subgroup of prime exponent p and a finite non-abelian p-subgroup.


Simple groups locally finite groups non-Abelian rank 

Mathematics Subject Classification




  1. 1.
    Brawley, J.V., Schnibben, G.E.: Infinite algebraic extensions of finite fields. American Mathematical Society, Providence (1989)CrossRefGoogle Scholar
  2. 2.
    Carter, R.W.: Simple groups of Lie type. Wiley, New York (1972)zbMATHGoogle Scholar
  3. 3.
    Dashkova, OYu.: Groups of finite non-Abelian sectional rank. Ukrainian Math. J. 49, 1494–1500 (1997)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ersoy, K., Kuzucuoǧlu, M.: Centralizers of subgroups in simple locally finite groups. J. Group Theory 15, 9–22 (2012)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hartley, B.: A general Brauer–Fowler theorem and centralizers in locally finite groups. Pacif. J. Math. 152, 101–117 (1992)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hartley, B., Kuzucuoǧlu, M.: Centralizers of elements in locally finite simple groups. Proc. London Math. Soc. 62, 301–324 (1991)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hartley, B., Shute, G.: Monomorphisms and direct limits of finite groups of Lie type. Quart. J. Math. Oxford 35, 49–71 (1984)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Huppert, B.: Endliche Gruppen I. Springer-Verlag, Berlin (1967)CrossRefGoogle Scholar
  9. 9.
    Kegel, O.H., Wehrfritz, B.A.F.: Locally finite groups. Elsevier, New York (1973)zbMATHGoogle Scholar
  10. 10.
    Kuzucuoǧlu, M.: Centralizers in simple locally finite groups. Int. J. Group Theory 2, 1–10 (2013)MathSciNetGoogle Scholar
  11. 11.
    Phillips, R.E.: Countably recognizable classes of groups. Rocky Mountain J. Math. 1, 489–497 (1971)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Robinson, D.J.S.: Finiteness conditions and generalized soluble groups. Springer, Berlin (1972)CrossRefGoogle Scholar
  13. 13.
    Šunkov, V.P.: On the theory of generalized solvable groups (in Russian). Dokl. Akud. Nuuk SSSR 160, 1279–1282 (1965)Google Scholar
  14. 14.
    Šuzuki, M.: On a class of doubly transitive groups. Ann. Math. 75, 105–145 (1962)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Ward, H.N.: On Ree’s series of simple groups. Trans. Am. Math. Soc. 121, 62–89 (1966)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Wehrfritz, B.A.F.: Infinite linear groups. Springer, Berlin (1973)CrossRefGoogle Scholar
  17. 17.
    Zaicev, D.I.: On solvable subgroups of locally solvable groups. Soviet Math. Dokl. 15, 342–345 (1974)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Dipartimento di Matematica e FisicaUniversità della Campania “Luigi Vanvitelli”CasertaItaly

Personalised recommendations