On self-similar finite $p$-groups

TL;DR

This paper characterizes when finite p-groups are self-similar, providing bounds on their order and a complete classification for groups of maximal class based on subgroup structure.

Contribution

It establishes bounds on the order of self-similar finite p-groups and fully characterizes those of maximal class in terms of subgroup properties.

Findings

Self-similar finite p-groups have bounded order depending on p and rank.

Groups of maximal class are self-similar iff they contain a specific elementary abelian subgroup.

The maximum order of such groups of maximal class is p^p+1, which is sharp.

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.