关于自相似有限$p$ -群
On self-similar finite $p$-groups
摘要：
在本文中，我们解决了以下问题：当是一个有限的$p$ -组$G$自相似，即当$G$可以忠实地表示为自同构的$p$ -进树的自相似自同构群？证明了若$G$是秩为$r$的自相似有限$p$ -群，则它的阶由$p$和$r$的函数有界.这特别适用于给定coclass的有限$p$ -群。在极大类群的特殊情况下，即在余类$1$的特殊情况下，我们可以完全回答上面的问题：极大类$G$的$p$ -群是自相似的当且仅当它包含一个初等阿贝尔极大子群，$G$在该子群上分裂。此外，在这种情况下，$G$的阶数至多为$p^p+1$，并且该界限是尖锐的。
英文摘要：
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.
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