The Gao-Zhuang conjecture for the Heisenberg group over 𝔽_p

Abstract

Let G be a finite nonabelian group. The small Davenport constant d(G) of G is the largest integer ℓ such that there exists a product-one free sequence over G of length ℓ, while the Gao constant E(G) of G is the least integer ℓ such that every sequence over G of length at least ℓ contains a product-one subsequence of length exactly |G|. A long-standing conjecture of Gao and Zhuang \cite{ZG2005} asserts that E(G)=d(G)+|G| for every finite nonabelian group G. Let p be an odd prime and let H_p³=UT₃(𝔽_p) be the finite Heisenberg group over 𝔽_p. Godara and Sarkar proved the Gao-Zhuang equality for H₂₇=UT₃(𝔽₃) and asked whether the same equality holds for H_p³ for every odd prime p. Recently, Volkmann proved that d(H_p³)=3p−3. In this paper, we determine the Gao constant of H_p³ and prove that E(H_p³)=d(H_p³)+|H_p³|=p³+3p−3. Together with the known abelian and cyclic-index cases, this completes the verification of the Gao-Zhuang equality for all groups of order p³, for every prime p.

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BibTeX RIS

BibTeX

@misc{qu2026gaozhuang,
  title = {{The Gao-Zhuang conjecture for the Heisenberg group over $\mathbbF_p$}},
  author = {Yongke Qu and Guoqing Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.23319},
  eprint = {2608.23319},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.23319},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - The Gao-Zhuang conjecture for the Heisenberg group over 𝔽_p
AU  - Yongke Qu
AU  - Guoqing Wang
PY  - 2026
DA  - 2026/09/06
PB  - arXiv
DO  - 10.48550/arxiv.2608.23319
UR  - https://arxiv.org/abs/2608.23319
AB  - Let G be a finite nonabelian group. The small Davenport constant d(G) of G is the largest integer ℓ such that there exists a product-one free sequence over G of length ℓ, while the Gao constant E(G) of G is the least integer ℓ such that every sequence over G of length at least ℓ contains a product-one subsequence of length exactly |G|. A long-standing conjecture of Gao and Zhuang \cite{ZG2005} asserts that E(G)=d(G)+|G| for every finite nonabelian group G. Let p be an odd prime and let H_p^3=UT_3(𝔽_p) be the finite Heisenberg group over 𝔽_p. Godara and Sarkar proved the Gao-Zhuang equality for H_27=UT_3(𝔽_3) and asked whether the same equality holds for H_p^3 for every odd prime p. Recently, Volkmann proved that d(H_p^3)=3p−3. In this paper, we determine the Gao constant of H_p^3 and prove that E(H_p^3)=d(H_p^3)+|H_p^3|=p^3+3p−3. Together with the known abelian and cyclic-index cases, this completes the verification of the Gao-Zhuang equality for all groups of order p^3, for every prime p.
ER  -

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