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Zbl 1069.20021
Campbell, C.M.; Cambell, P.P.; Doostie, H.; Robertson, E.F.
Fibonacci lengths for certain metacyclic groups. (English)
[J] Algebra Colloq. 11, No. 2, 215-222 (2004). ISSN 1005-3867

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 $F_A(G)$, and the Fibonacci length of $G$ with respect to $A$, denoted by $\text{LEN}_A(G)$ or $\text{LEN}(G)$, are defined in the usual manner.\par In this paper, the authors examine the Fibonacci length of certain classes of 2-generator metacyclic groups including the metacyclic Fox groups $G_{n,l}=\langle a,b\mid ab^n=b^la$, $ba^n=a^lb\rangle$. They also study the Fibonacci length of the groups $F(r,2)$ when $r$ is odd. They prove that $\text{LEN}(F(3,2))=3$ and find $\text{LEN}(F(r,2))$ when $r\ge 5$. In case $r$ is even the length is known by a result due to D. D. Wall (1960).
[C. G. Chehata (Orlando)]

MSC 2000:: *20F05 Presentations of groups
20F16 Solvable groups
11B39 Special numbers, etc.

Keywords: Fibonacci orbit; Fibonacci lengths of groups; 2-generator metacyclic groups

Citations: Zbl 0101.03201

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