the $n$-ary adding machine and solvable groups

نویسندگان

josimar da silva rocha

instituto federal de educacao said sidki

universidade de brasilia

چکیده

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-1right) $‎. ‎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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عنوان ژورنال:
international journal of group theory

جلد ۲، شماره ۴، صفحات ۴۳-۸۸

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