Motivated by a question about
geometric packings in n-dimensional Euclidean space, Rn, we consider the
following problem about finite abelian groups. Let n be an integer, n ≥ 3, and
let k be a positive integer. Let g(k,n) be the order of the smallest abelian
group in which there exist n elements, a1,a2,…,an, such that the kn elements
iaj, 1 ≤ i ≤ k, are distinct and not 0. We will show that for n fixed,
g(k,n) ∼ 2cos(π∕n)k3∕2.
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