Exponent of self-similar finite \(p\)-groups (Q6619332)

A group \(G\) is self-similar of some integer degree \(m\) if it has a faithful representation on an infinite regular one-rooted \(m\)-tree \(\mathcal{T}_{m}\) such that the representation is state-closed and is transitive on the tree's first level.\N\NThe main result in the paper under review is Theorem 1.1: Let \(p\) be a prime and \(G\) a pro-\(p\) group of finite rank such that \(\{ g \in G \mid g^{p^{n}}=1 \}\) is a non-trivial subgroup of \(G\) for some \(n\). If \(G\) is self-similar of degree \(p\), then \(G\) is a finite \(p\)-group and has exponent at most \(p^{n}\).\N\NThanks to Theorem 1.1, the authors obtain the following generalization of a result by \textit{Z. Šunić} [Int. J. Algebra Comput. 21, No. 1--2, 355--364 (2011; Zbl 1233.20019)]. \N\NTheorem 1.2: Let \(p\) be a prime. If \(G\) is a self-similar finite $p$-group of degree \(p\) such that the first level stabilizer is power abelian, then \(G\) is a split extension of a \(p\)-group of exponent \(p\) by a cyclic group of order \(p\).