On self-similar finite -groups

  • Azam Babai

    University of Qom, Iran
  • Khadijeh Fathalikhani

    University of Kashan, Iran
  • Gustavo A. Fernández-Alcober

    Universidad del Pais Vasco, Bilbao, Spain
  • Matteo Vannacci

    Universidad del Pais Vasco, Bilbao, Spain
On self-similar finite $p$-groups cover

A subscription is required to access this article.


In this paper, we address the following question: when is a finite -group self-similar, i.e. when can be faithfully represented as a self-similar group of automorphisms of the -adic tree? We show that, if is a self-similar finite -group of rank , then its order is bounded by a function of and . This applies in particular to finite -groups of a given coclass. In the particular case of groups of maximal class, that is, of coclass 1, we can fully answer the question above: a -group of maximal class is self-similar if and only if it contains an elementary abelian maximal subgroup over which splits. Furthermore, in that case the order of is at most , and this bound is sharp.

Cite this article

Azam Babai, Khadijeh Fathalikhani, Gustavo A. Fernández-Alcober, Matteo Vannacci, On self-similar finite -groups. Groups Geom. Dyn. 14 (2020), no. 2, pp. 337–348

DOI 10.4171/GGD/546