How long does it take to generate a group?

TL;DR

This paper investigates the diameter of finite abelian groups relative to various generating sets, identifying maximum diameters, classifying extremal generating sets, and exploring related combinatorial structures.

Contribution

It determines the maximum possible diameter for finite abelian groups and classifies all generating sets achieving this maximum, also analyzing size constraints and related combinatorial concepts.

Findings

Maximum diameter values for finite abelian groups identified

Classification of generating sets attaining maximum diameter provided

Connections with caps, sum-free sets, and quasi-perfect codes discussed

Abstract

The diameter of a finite group $G$ with respect to a generating set $A$ is the smallest non-negative integer $n$ such that every element of $G$ can be written as a product of at most $n$ elements of $A \cup A^{-1}$. We denote this invariant by $\diam_A(G)$. It can be interpreted as the diameter of the Cayley graph induced by $A$ on $G$ and arises, for instance, in the context of efficient communication networks. In this paper we study the diameters of a finite abelian group $G$ with respect to its various generating sets $A$. We determine the maximum possible value of $\diam_A(G)$ and classify all generating sets for which this maximum value is attained. Also, we determine the maximum possible cardinality of $A$ subject to the condition that $\diam_A(G)$ is "not too small". Connections with caps, sum-free sets, and quasi-perfect codes are discussed.