Finite self-similar p-groups with abelian first level stabilizers

Abstract

We determine all finite p-groups that admit a faithful, self-similar action on the p-ary rooted tree such that the first level stabilizer is abelian. A group is in this class if and only if it is a split extension of an elementary abelian p-group by a cyclic group of order p. The proof is based on use of virtual endomorphisms. In this context the result says that if G is a finite p-group with abelian subgroup H of index p, then there exists a virtual endomorphism of G with trivial core and domain H if and only if G is a split extension of H and H is an elementary abelian p-group.

one direction of theorem 2 extended to regular p-groups

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Cited by in corpus (2)

• On self-similarity of $p$-adic analytic pro-$p$ groups of small dimension
• On the self-similarity index of $p$-adic analytic pro-$p$ groups