How long does it take to generate a group?

Abstract

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 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.