How long does it take to generate a group?

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 A−1. We denote this invariant by diamA(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 diamA(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 diamA(G) is “not too small”. Connections with caps, sum-free sets, and quasi-perfect codes are discussed.