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.

中文翻译：

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在自相似有限$ p $ -groups上

在本文中，我们解决以下问题：什么时候有限的$ p $ -group $ G $自相似，即何时$ G $能够忠实地表示为$ p $ -adic自同构的自相似组树？我们证明，如果$ G $是等级为rr $的自相似有限$ p $-组，则其阶数受$ p $和$ r $的函数限制。这尤其适用于给定共类的有限$ p $ -groups。在最大类的组，即协类1的特殊情况下，我们可以完全回答上面的问题：最大类$ G $的$ p $-组是自相似的，当且仅当它包含基本阿贝尔语$ G $拆分的最大子组。此外，在这种情况下，$ G $的顺序最多为$ p ^ {p + 1} $，这个界限很明显。