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Vietnam Journal of Mathematics 33:2 (2005) 189-197

Fibonacci Length of Direct Products of Groups

H. Doostie and M. Maghasedi

Abstract. For a non-abelian finite group $G=\langle a_1,a_2,\dots ,a_n\rangle$ the Fibonacci length of $G$ with respect to the ordered generating set $A=\{ a_1,a_2,\dots ,a_n \} $ is the least integer $l$ such that for the sequence of elements $x_i=a_i,~1\leq i\leq n,~ x_{n+i}=\prod_{j=1}^{n}x_{i+j-1},~i\geq 1$, of $G$, the equations $x_{l+i}=a_i,~1\leq i\leq n$ hold. The question posed in $2003$ by P. P. Campbell that "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.