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 $$r(G\oplus G')\leq r(G)+4$$.

MSC:

 20D60 Arithmetic and combinatorial problems involving abstract finite groups 20K01 Finite abelian groups

Zbl 0245.20046
Full Text:

References:

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