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.