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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da Silva Rocha, J., & Sidki, S. (2013). The $n$-ary adding machine and solvable groups. International Journal of Group Theory, 2(4), 43-88. doi: 10.22108/ijgt.2013.2871
MLA
Josimar da Silva Rocha; Said Sidki. "The $n$-ary adding machine and solvable groups". International Journal of Group Theory, 2, 4, 2013, 43-88. doi: 10.22108/ijgt.2013.2871
HARVARD
da Silva Rocha, J., Sidki, S. (2013). 'The $n$-ary adding machine and solvable groups', International Journal of Group Theory, 2(4), pp. 43-88. doi: 10.22108/ijgt.2013.2871
VANCOUVER
da Silva Rocha, J., Sidki, S. The $n$-ary adding machine and solvable groups. International Journal of Group Theory, 2013; 2(4): 43-88. doi: 10.22108/ijgt.2013.2871
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