# On self-similar finite $p$-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

## Abstract

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

## Cite this article

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

DOI 10.4171/GGD/546