Mathematics – Group Theory
Scientific paper
2009-11-15
J. Algebra 261 (2003), no. 1, 145--171
Mathematics
Group Theory
Scientific paper
10.1016/S0021-8693(02)00671-3
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.
Klopsch Benjamin
Lev Vsevolod F.
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