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Property D

Property D for a group G=(Z/nZ)^r, where n>=2 and r>=1, is the assertion that every sequence of length s(G)-1 having no zero-sum subsequence of exactly n terms has the form T^(n-1). Here s(G) is the least length forcing such a subsequence, and T^(n-1) denotes n-1 repetitions of one sequence T (Gao et al. 2007, p. 246).

In this form, T has no repeated terms. Equivalently, every element in the extremal sequence occurs exactly n-1 times. The assertion concerns sequences of maximum possible length, not merely ones that cannot be extended.

Wang and Zeng (2026) proved that, for every r>=4 and even n>=2, this property holds iff n is a power of 2, refuting its proposed validity for all n and r. Their construction also improves Erdős-Ginzburg-Ziv constants.

See also

Property C, Zero-Sum Sequence

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References

Gao, W. and Geroldinger, A. "Zero-Sum Problems in Finite Abelian Groups: A Survey." Expo. Math. 24, 337-369, 2006. https://doi.org/10.1016/j.exmath.2006.07.002.Gao, W.; Geroldinger, A.; and Schmid, W. A. "Inverse Zero-Sum Problems." Acta Arith. 128, 245-279, 2007. https://doi.org/10.4064/aa128-3-5.Wang, G. and Zeng, X. "Inverse Zero-Sum Problems of Erdős-Ginzburg-Ziv Constant: Property C and Property D Conjectures." 25 Oct 2026. https://arxiv.org/abs/2610.25271.

Cite this as:

Weisstein, Eric W. "Property D." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PropertyD.html

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