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Davenport Constant


The Davenport constant of a finite Abelian group G is defined to be the length of the longest minimal zero-system of G and is denoted D(G). Symbolically,

 D(G)=max{|sigma|:sigma in U(G)}.

D({0})=1 for completeness.

In order words, if G is a finite Abelian group of order n, then the Davenport constant of G is the minimal d such that every sequence of elements of G with length d contains a nonempty subsequence with a zero-sum.

Some values of the Davenport constant include the following.

1. D(Z_n)=n.

2. Let G= direct sum _(i=1)^rZ_(p^(e_i)) be a finite p-group. Then D(G)=1+sum_(i=1)^(r)(p^(e_i)-1).

3. Let G=Z_n direct sum Z_m with m|n. Then D(G)=m+n-1

One open question in finite group theory is the determination of a general formula for D(G).


This entry contributed by Nick Hutzler

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References

Chapman, S. T. "On the Davenport Constant, the Cross Number, and Their Applications in Factorization Theory." In Zero-Dimensional Commutative Rings (Ed. D. F. Anderson and D. E. Dobbs). New York: Dekker, pp. 167-190, 1997.

Referenced on Wolfram|Alpha

Davenport Constant

Cite this as:

Hutzler, Nick. "Davenport Constant." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/DavenportConstant.html

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