Finite generation of iterated wreath products

2010/02/01 by Ievgen Bondarenko, Bondarenko, Ievgen · 2 citations
Computer Science · Mathematics · #20E08 #20E18 #20E22 #20F05 #Advanced Topology and Set Theory #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #math.GR #msc:20E08 #msc:20E18 #msc:20E22 #msc:20F05 #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1002.0320

7 pages. An example of a branch group with maximal subgroups of infinite index is added

openalex publication_date 2010/02/01 · arxiv created 2010/07/01 · arxiv updated 2010/07/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ...\wr G₂\wr G₁ is topologically finitely generated if and only if the profinite abelian group ∏_{n≥ 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...\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, answering [2,Question 14].

Cited by

• Generation of Iterated Wreath Products Constructed from Almost Simple Groups
• Branch groups with many maximal subgroups

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