Generating abelian groups by addition only

TL;DR

This paper investigates the maximum positive diameter of finite abelian groups with respect to their generating sets, classifies the sets achieving this maximum, and explores bounds related to the size of these sets.

Contribution

It determines the maximum positive diameter for finite abelian groups, classifies the extremal generating sets, and bounds the size of generating sets with large diameter.

Findings

Identified the maximum positive diameter for finite abelian groups.

Classified all generating sets that attain this maximum.

Bounded the size of generating sets with large positive diameter.

Abstract

We define the positive diameter of a finite group $G$ with respect to a generating set $A\subset G$ to be the smallest non-negative integer $n$ such that every element of $G$ can be written as a product of at most $n$ elements of $A$. This invariant, which we denote by $\diam_A^+(G)$, can be interpreted as the diameter of the Cayley digraph induced by $A$ on $G$. In this paper we study the positive diameters of a finite abelian group $G$ with respect to its various generating sets $A$. More specifically, 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". Conceptually, the problems studied are closely related to our earlier work and the results obtained shed a new light on the subject. Our original motivation came from connections with caps, sum-free sets, and quasi-perfect codes.