Fibonacci sequences in finite groups.

*(English)*Zbl 0758.20006A $k$-nacci sequence in a finite group $G$ is a sequence of group elements ${x}_{0},{x}_{1},\cdots ,{x}_{n},\cdots $ for which, given an initial generating set ${x}_{0},\cdots ,{x}_{j-1}$ for $G$, each element is defined by

$${x}_{n}=\left\{\begin{array}{cc}{x}_{0}{x}_{1}\cdots {x}_{n-1}\hfill & \text{for}\phantom{\rule{4.pt}{0ex}}j\le n<k\hfill \\ {x}_{n-k}{x}_{n-k+1}\cdots {x}_{n-1}\hfill & \text{for}\phantom{\rule{4.pt}{0ex}}n\ge k\xb7\hfill \end{array}\right.$$

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.

Reviewer: G.Rosenberger (Dortmund)

##### MSC:

20D60 | Arithmetic and combinatorial problems on finite groups |

11B39 | Fibonacci and Lucas numbers, etc. |

20F05 | Generators, relations, and presentations of groups |