Upper Bounds For The Diameter Of A Direct Power Of Solvable Groups

Abstract: Let G be a finite group with a generating set A. By the (symmetric) diameter of G with respect to A we mean the maximum over g in G of the length of the shortest word in (A union A inverse)A expressing g.By the (symmetric) diameter of G we mean the maximum of (symmetric) diameter over all generating sets of G. Let n greater than or equal to 1, by G power n we mean the n-th direct power of G. For n greater than or equal to 1 and finite non-abelian solvable group G we find an upper bound, growing polynomially with respect to n, for the symmetric diameter and the diameter of G power n.