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Fibonacci sequences in finite groups. (English) Zbl 0758.20006

A k-nacci sequence in a finite group G is a sequence of group elements x 0 ,x 1 ,,x n , for which, given an initial generating set x 0 ,,x j-1 for G, each element is defined by

x n =x 0 x 1 x n-1 forjn<kx n-k x n-k+1 x n-1 fornk·

A k-nacci sequence certainly reflects the structure of G. A finite group G is called k-nacci sequenceable if there exists a k-nacci sequence of G such that every element of G appears in the sequence. It is shown that a k-nacci sequence in a finite group G is simply periodic. This leads to a complete description of the 2-nacci sequenceable groups. A 2-nacci sequenceable group is cyclic.

MSC:
20D60Arithmetic and combinatorial problems on finite groups
11B39Fibonacci and Lucas numbers, etc.
20F05Generators, relations, and presentations of groups