Archiv der Mathematik

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

Finite generation of iterated wreath products

  • Ievgen V. BondarenkoEmail author


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 


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Copyright information

© Springer Basel AG 2010

Authors and Affiliations

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

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