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Fibonacci lengths for certain metacyclic groups. (English) Zbl 1069.20021

The authors consider finitely generated groups. If $A$ is a finite generating system of the group $G$, then the Fibonacci orbit of $G$ with respect to $A$, denoted by ${F}_{A}\left(G\right)$, and the Fibonacci length of $G$ with respect to $A$, denoted by ${\text{LEN}}_{A}\left(G\right)$ or $\text{LEN}\left(G\right)$, are defined in the usual manner.

In this paper, the authors examine the Fibonacci length of certain classes of 2-generator metacyclic groups including the metacyclic Fox groups ${G}_{n,l}=〈a,b\mid a{b}^{n}={b}^{l}a$, $b{a}^{n}={a}^{l}b〉$. They also study the Fibonacci length of the groups $F\left(r,2\right)$ when $r$ is odd. They prove that $\text{LEN}\left(F\left(3,2\right)\right)=3$ and find $\text{LEN}\left(F\left(r,2\right)\right)$ when $r\ge 5$. In case $r$ is even the length is known by a result due to D. D. Wall (1960).

##### MSC:
 20F05 Generators, relations, and presentations of groups 20F16 Solvable groups, supersolvable groups 11B39 Fibonacci and Lucas numbers, etc.