Bound on the diameter of split metacyclic groups

Abstract

Let $G_{m,n,k} = \mathbb{Z}_m \ltimes_k \mathbb{Z}_n$ 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 \textit{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.