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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 (G), and the Fibonacci length of G with respect to A, denoted by LEN A (G) or LEN(G), 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,bab n =b l a, ba n =a l b. They also study the Fibonacci length of the groups F(r,2) when r is odd. They prove that LEN(F(3,2))=3 and find LEN(F(r,2)) when r5. In case r is even the length is known by a result due to D. D. Wall (1960).

MSC:
20F05Generators, relations, and presentations of groups
20F16Solvable groups, supersolvable groups
11B39Fibonacci and Lucas numbers, etc.