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Fibonacci length of generating pairs in groups. (English) Zbl 0741.20025
Applications of Fibonacci numbers. Vol. 3, Proc. 3rd Int. Conf., Pisa/Italy 1988, 27-35 (1990).
[For the entire collection see Zbl 0699.00017.] For a 2-generator group $G$ (generated by $x$ and $y$), the authors define the Fibonacci orbit $F\sb{x,y}=\{a\sb i\}$ of $(x,y)$ by $a\sb 1=x$, $a\sb 2=y$, $a\sb{i+2}=a\sb i\cdot a\sb{i+1}$, $i\ge 1$ and the basic Fibonacci orbit $\bar F\sb{x,y}$ of basic length $m$ by the sequence $\{a\sb i\}$ of elements of $G$ such that $a\sb 1=x$, $a\sb 2=y$, $a\sb{i+2}=a\sb i\cdot a\sb{i+1}$, $i\ge 1$, where $m\ge 1$ is the least integer with $a\sb 1=a\sb{m+1}\theta$, $a\sb 2=a\sb{m+2}\theta$, for some $\theta\in\hbox{Aut}(G)$. Also, the Fibonacci length of a generator pair $(x,y)$ (LEN) is defined by the number of elements of $F\sb{x,y}$ and the length of the basic Fibonacci orbit (BLEN) by the number of elements of $\bar F\sb{x,y}$. The main aim of this paper is to compute LEN and BLEN for many classical finite groups: $D\sb{2n}$, $Q\sb 8$ and $PSL(2,p)$ ($p$ prime number).

20F05Generators, relations, and presentations of groups