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Archiv der Mathematik

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

Finite generation of iterated wreath products

  • Ievgen V. Bondarenko
Article

Abstract

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 

Keywords

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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