Generating abelian groups by addition only

Abstract

We define the positive diameter of a finite group G with respect to a generating set A ⊆ 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 , 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 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 is “not too small”. Conceptually, the problems studied are closely related to our earlier work [Klopsch and Lev, J. Algebra 261: 145–171, 2003] 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.