  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 $A \cup A^{-1}$. We denote this
invariant by $\diam_A(G)$. 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 $\diam_A(G)$ 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". Connections with
caps, sum-free sets, and quasi-perfect codes are discussed.
