A subgroup involvement of the Fibonacci length

For a non-Abelian 2-generated finite group G =〈 a , b 〉, the Fibonacci length of G with respect to A ={ a , b }, denoted by LEN A ( G ), is defined to be the period of the sequence x 1 = a , x 2 = b , x 3 = x 1 x 2 ,…, x n +1 = x n −1 x n ,… of the elements of G . For a finite cyclic group C n =〈 a 〉, LEN A ( C n ) is defined in a similar way where A ={1, a } and it is known that LEN A ( C n )= k ( n ), the well-known Wall number of n . Over all of the interesting numerical results on the Fibonacci length of finite groups which have been obtained by many authors since 1990, an intrinsic property has been studied in this paper. Indeed, by studying the family of minimal non-Abelian p-groups it will be shown that for every group G of this family, there exists a suitable generating set A ′ for the derived subgroup G ′ such that LEN A ′ ( G ′)| LEN A ( G ) where, A is the original generating set of G .