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

TL;DR

This paper classifies finite p-groups that act faithfully and self-similarly on p-ary rooted trees with abelian first level stabilizers, showing they are split extensions of elementary abelian p-groups by cyclic groups of order p.

Contribution

It provides a complete characterization of such p-groups using virtual endomorphisms, establishing a precise structural criterion for their existence.

Findings

Finite p-groups with abelian first level stabilizers are split extensions of elementary abelian p-groups by cyclic groups of order p.

Existence of a virtual endomorphism with trivial core characterizes these groups.

The classification is based on the use of virtual endomorphisms and group extension theory.

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.