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

TL;DR

This paper establishes polynomial upper bounds on the diameter of direct powers of finite non-abelian solvable groups, advancing understanding of their algebraic structure and growth properties.

Contribution

It provides the first polynomial bounds for the diameter of direct powers of finite non-abelian solvable groups, a significant step in group theory.

Findings

Polynomial upper bounds for diameters of G^n

Bounds grow polynomially with n

Applicable to non-abelian 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.