The n-ary Adding Machine and Soluble Groups

TL;DR

This paper investigates the structure of abelian subgroups in the automorphism group of an n-ary tree, focusing on those normalized by the n-ary adding machine, and characterizes soluble subgroups for prime n and n=4.

Contribution

It provides a detailed description of abelian subgroups normalized by the n-ary adding machine and classifies soluble subgroups containing it for specific n values.

Findings

Abelian subgroups normalized by the adding machine are characterized under various conditions.

For prime n and n=4, soluble subgroups containing the adding machine are classified as extensions of specific types.

The structure of soluble subgroups is explicitly described in these cases.

Abstract

We describe under a variety of conditions abelian subgroups of the automorphism group A of the regular n-ary tree T which are normalized by the n-ary adding machine t=(e,...,e,t)s where s is the n-cycle (0,1,...,n-1). As an application, for n a prime number, and for n = 4 we prove that every soluble subgroup of A containing t is an extension of a torsion-free metabelian group by a finite group.