Upper bounds for the diameter of a direct power of non-abelian solvable groups

Consider G as a finite group with a generating set A. We define the (symmetric) diameter of G with respect to A as the maximum length of the shortest word in ( \(A \cup A^{-1}) A\) $A\cup A^{-1})A$ expressing g, where g ranges over all elements in G. The (symmetric) diameter of G is then the maximum (symmetric) diameter over all possible generating sets of G. We use \(G^n\) $G^n$ to denote the n-th direct power of G. For any finite non-abelian solvable group G and \(n\ge 1\) $n\ge 1$ , we establish an upper bound for both the symmetric diameter and the diameter of \(G^n\) $G^n$ . This upper bound grows polynomially with respect to n.