## 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

### Keywords:

minimal zero sequences
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