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On long minimal zero sequences in finite abelian groups. (English) Zbl 0980.11014

From the introduction: This paper centres around the following problem: let \(G\) be a finite abelian group and \(D(G)\) Davenport’s constant of \(G\). Consider a long minimal zero sequence resp. a long zero-free sequence \(S\); where in this context long means that \(D(G)-|S|\) is small. What can be said about the structure of \(S\)? There are simple, well-known answers for cyclic groups and elementary 2-groups. Our aim is to derive similar results for more general groups. We study the action of the automorphism group, and we ask for the order of elements in \(S\). If the rank of \(G\) is large, then all elements of \(S\) may be pairwise distinct. Conversely, if the exponent is large, then one element occurs with high multiplicity. We develop a polynomial method which can be applied successfully to elementary \(p\)-groups.

MSC:

11B75 Other combinatorial number theory
20K01 Finite abelian groups
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