Bound on the diameter of split metacyclic groups

TL;DR

This paper establishes an upper bound on the diameter of split metacyclic groups using an arithmetic parameter called weight, with implications for understanding the diameter of broader metacyclic groups.

Contribution

It introduces a novel upper bound for the diameter of split metacyclic groups based on the weight parameter, extending to general metacyclic groups.

Findings

Derived an explicit upper bound for the diameter of $G_{m,n,k}$

Connected the diameter bounds to the weight parameter involving $n$, $k$, and the order of $k$

Provided a method to estimate diameters of arbitrary 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.