Bound on the diameter of metacyclic groups

Abstract

Let G m,n,k = Zm ⋉ k Zn be the split metacyclic group, where k is a unit modulo n. We derive an upper bound for the diameter of G m,n,k using an arithmetic parameter called the weight, which depends on n, k, and the order of k. As an application, we show how this would determine a bound on the diameter of an arbitrary metacyclic group. t i=1x ai y bi = x a1+...+at y b1k a 2 +...+a t +...+bt−1k a t +bt .