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PreprintCC BY 4.0arXiv

The Erdős--Ginzburg--Ziv theorem for finite groups

  • Yongke Qu
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Abstract

Let G be a finite group, and let p be the smallest prime divisor of |G|. The Gao constant E(G) is the least integer ℓ such that every sequence over G of length ℓ contains a subsequence of length |G| whose terms can be ordered to have product equal to the identity. In 1961, Erdős, Ginzburg and Ziv proved that every sequence of 2n−1 integers has an n-term subsequence whose sum is divisible by n; equivalently, E(Cn)≤ 2n−1, where Cn denotes the cyclic group of order n. In this paper, we prove that E(G)≤ |G|+|G|/p−1 whenever G contains no cyclic subgroup of index p. Consequently, every finite non-cyclic group satisfies E(G)≤ |G|+|G|/p+p−2, with equality if and only if G contains a cyclic subgroup of index p. This establishes the conjectured upper bound for the Gao constant and provides a complete characterization of the equality case.

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Cite

BibTeXRIS
BibTeX
@misc{qu2026erdősginzburgziv,
  title = {{The Erdős--Ginzburg--Ziv theorem for finite groups}},
  author = {Yongke Qu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.26122},
  eprint = {2610.26122},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.26122},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The Erdős--Ginzburg--Ziv theorem for finite groups
AU  - Yongke Qu
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.26122
UR  - https://arxiv.org/abs/2610.26122
AB  - Let G be a finite group, and let p be the smallest prime divisor of |G|. The Gao constant E(G) is the least integer ℓ such that every sequence over G of length ℓ contains a subsequence of length |G| whose terms can be ordered to have product equal to the identity. In 1961, Erdős, Ginzburg and Ziv proved that every sequence of 2n−1 integers has an n-term subsequence whose sum is divisible by n; equivalently, E(C_n)≤ 2n−1, where C_n denotes the cyclic group of order n. In this paper, we prove that E(G)≤ |G|+|G|/p−1 whenever G contains no cyclic subgroup of index p. Consequently, every finite non-cyclic group satisfies E(G)≤ |G|+|G|/p+p−2, with equality if and only if G contains a cyclic subgroup of index p. This establishes the conjectured upper bound for the Gao constant and provides a complete characterization of the equality case.
ER  -