JournalsggdVol. 14, No. 2pp. 337–348

On self-similar finite pp-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
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Abstract

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

Cite this article

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

DOI 10.4171/GGD/546