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How long does it take to generate a group?
The diameter of a finite group with respect to a generating set is
the smallest non-negative integer such that every element of can be
written as a product of at most elements of . We denote this
invariant by \diam_A(G). It can be interpreted as the diameter of the Cayley
graph induced by on and arises, for instance, in the context of
efficient communication networks.
In this paper we study the diameters of a finite abelian group with
respect to its various generating sets . 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
subject to the condition that \diam_A(G) is "not too small". Connections with
caps, sum-free sets, and quasi-perfect codes are discussed
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