On self-similar finite \(p\)-groups (Q784884)

Summary: 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.