Fibonacci length of direct products of groups.

*(English)* Zbl 1123.20029
Summary: For a non-Abelian finite group $G=\langle {a}_{1},{a}_{2},\cdots ,{a}_{n}\rangle $ the Fibonacci length of $G$ with respect to the ordered generating set $A=\{{a}_{1},{a}_{2},\cdots ,{a}_{n}\}$ is the least integer $l$ such that for the sequence of elements ${x}_{i}={a}_{i}$, $1\le i\le n$, ${x}_{n+i}={\prod}_{j=1}^{n}{x}_{i+j-1}$, $i\ge 1$, of $G$, the equations ${x}_{l+i}={a}_{i}$, $1\le i\le n$ hold. The question posed in 2003 by P. P. Campbell is “Is there any relationship between the lengths of finite groups $G$, $H$ and $G\times H$?” In this paper we answer this question when at least one of the groups is a non-Abelian 2-generated group.

##### MSC:

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

20D60 | Arithmetic and combinatorial problems on finite groups |

11B39 | Fibonacci and Lucas numbers, etc. |

20D40 | Products of subgroups of finite groups |