Archiv der Mathematik

, Volume 95, Issue 4, pp 301–308 | Cite as

Finite generation of iterated wreath products

  • Ievgen V. Bondarenko


Let (G n , X n ) be a sequence of finite transitive permutation groups with uniformly bounded number of generators. We prove that the infinitely iterated permutational wreath product \({\ldots\wr G_2\wr G_1}\) is topologically finitely generated if and only if the profinite abelian group \({\prod_{n\geq 1} G_n/G'_n}\) is topologically finitely generated. As a corollary, for a finite transitive group G the minimal number of generators of the wreath power \({G\wr \ldots\wr G\wr G}\) (n times) is bounded if G is perfect, and grows linearly if G is non-perfect. As a by-product we construct a finitely generated branch group, which has maximal subgroups of infinite index.

Mathematics Subject Classification (2000)

Primary 20F05 20E22 Secondary 20E18 20E08 


Iterated wreath product Profinite group Inverse limit Branch group 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bartholdi L.: A Wilson group of non-uniformly exponential growth. C. R. Math. Acad. Sci. Paris 336, 549–554 (2003)MATHMathSciNetGoogle Scholar
  2. 2.
    L. Bartholdi, R. Grigorchuk, and Z. Šuniḱ, Branch groups, in: Handbook of algebra, vol. 3, pp. 989–1112. North-Holland, Amsterdam, 2003.Google Scholar
  3. 3.
    Bhattacharjee M.: The probability of generating certain profinite groups by two elements. Isr. J. Math. 86, 311–329 (1994)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    David C.: Generating numbers for wreath products. Rend. Semin. Mat. Univ. Padova 92, 71–77 (1994)MATHGoogle Scholar
  5. 5.
    Grigorchuk R.: Branch groups. Math. Notes 67, 718–723 (2000)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    R. Grigorchuk, Solved and unsolved problems around one group, in: Infinite groups: geometric, combinatorial and dynamical aspects, Progr. Math. 248, pp. 117–218. Birkhäuser, Basel, 2005.Google Scholar
  7. 7.
    Lucchini A.: Generating wreath products and their augmentation ideals. Rend. Semin. Mat. Univ. Padova 98, 67–87 (1997)MATHMathSciNetGoogle Scholar
  8. 8.
    Neumann P.: Some questions of Edjvet and Pride about infinite groups. Ill. J. Math. 30, 301–316 (1986)MATHGoogle Scholar
  9. 9.
    Quick M.: Probabilistic generation of wreath products of non-Abelian finite simple groups. Commun. Algebra 32, 4753–4768 (2004)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Quick M.: Probabilistic generation of wreath products of non-Abelian finite simple groups. II. Int. J. Algebra Comput. 16, 493–503 (2006)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Segal D.: The finite images of finitely generated groups. Proc. Lond. Math. Soc. (3) 82, 597–613 (2001)MATHCrossRefGoogle Scholar
  12. 12.
    Wiegold J.: Growth sequences of finite groups II. J. Austral. Math. Soc. 20, 225–229 (1975)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Wiegold J., Wilson J.: Growth sequences of finitely generated groups. Arch. Math. 30, 337–343 (1978)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Woryna A.: On generation of wreath products of cyclic groups by two state time varying Mealy automata. Int. J. Algebra Comput. 16, 397–415 (2006)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Mechanics and Mathematics DepartmentNational Taras Shevchenko University of KyivKievUkraine

Personalised recommendations