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On Davenport’s constant. (English) Zbl 0759.20008
For an additively written finite abelian group \(G\) Davenport’s constant \(D(G)\) is defined as the maximal length \(d\) of a sequence \((g_ 1,\dots,g_ d)\) in \(G\) such that \(g_ 1+\dots +g_ d=0\), but no finite subsum equals zero. If \(C_ n=\mathbb{Z}/n\mathbb{Z}\) and \(G=C_{n_ 1}\oplus\dots\oplus C_{n_ r}\), where \(1<n_ 1| n_ 2|\dots| n_ r\), then \(D(G)\geq M(G)=n_ 1+\cdots + n_ r-r+1\); here equality holds for several classes of groups, i.e. for all \(p\)- groups and all groups of rank \(r\leq 2\), but not for all abelian groups [P. van Emde Boas, D. Kruyswijk, A combinatorial problem on finite abelian groups III, Report ZW-1969-008 Math. Centrum Amsterdam (1969; Zbl 0245.20046)].
In the present paper, the authors provide several series of abelian groups \(G\) of rank \(r\geq 4\) satisfying \(D(G)>M(G)\). They also show that this phenomenon is not an exceptional one but appears rather frequently; one of the results in this direction is the following one (Corollary 1): For every finite abelian group \(G\) there exists a finite abelian group \(G'\) such that \(D(G\oplus G')>M(G\oplus G')\) and the ranks satisfy \(r(G\oplus G')\leq r(G)+4\).

20D60 Arithmetic and combinatorial problems involving abstract finite groups
20K01 Finite abelian groups
Full Text: DOI
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