Generating abelian groups by addition only

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.