The $n$-ary adding machine and solvable groups

Document Type: Research Paper

Authors

1 Instituto Federal de Educacao

2 Universidade De Brasilia

Abstract

We describe under various conditions abelian subgroups of the automorphism‎ ‎group $\mathrm{Aut}(T_{n})$ of the regular $n$-ary tree $T_{n}$‎, ‎which are‎ ‎normalized by the $n$-ary adding machine $\tau =(e‎, ‎\dots‎, ‎e,\tau )\sigma _{\tau‎ ‎}$ where $\sigma _{\tau }$ is the $n$-cycle $\left( 0,1‎, ‎\dots‎, ‎n-1\right) $‎. ‎As‎ ‎an application‎, ‎for $n=p$ a prime number‎, ‎and for $n=4$‎, ‎we prove that‎ ‎every soluble subgroup of $\mathrm{Aut}(T_{n})$‎, ‎containing $\tau $ is an extension of a torsion-free metabelian group by a‎ ‎finite group‎.

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