The Gao-Zhuang conjecture for the Heisenberg group over $\mathbb{F}_p$

Guoqing Wang; Yongke Qu

The Gao-Zhuang conjecture for the Heisenberg group over $\mathbb{F}_p$

T0 review · 3 major / 4 minor · reviewed 2026-08-28 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves that the Gao constant of $H_{p^3}$, the exponent-$p$ Heisenberg group of order $p^3$, is $p^3+3p-3$, confirming the Zhuang-Gao equality for every odd prime $p$.

desk verdict A genuine new theorem: E(H_{p^3}) = p^3 + 3p - 3 for every odd prime p; the proof is coherent and mostly self-contained, but one key lemma is imported from an unverified preprint and everything rests on it. read the letter →

arxiv 2608.23319 v2 pith:6OF7WOSE submitted 2026-08-24 math.CO math.NT

Yongke Qu , Guoqing Wang This is my paper · ORCID

classification math.COmath.NT MSC 11B7520D60

keywords GaoconstantsmallDavenportproduct-onesequencezero-sumHeisenberggroupZhuang-Gaoconjectureextraspecialp-group

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves the Zhuang-Gao conjecture for the exponent-$p$ Heisenberg group $H_{p^3} = \mathrm{UT}_3(\mathbb{F}_p)$ for every odd prime $p$. The claim is that the Gao constant equals the small Davenport constant plus the group order: $E(H_{p^3}) = d(H_{p^3}) + |H_{p^3}| = p^3 + 3p - 3$. Equivalently, any sequence of $p^3+3p-3$ elements of the group must contain a subsequence of exactly $p^3$ elements whose terms can be ordered to multiply to the identity, and this threshold is minimal. This matters because the equality $E(G)=d(G)+|G|$ is a classical theorem for abelian groups but is open in general for nonabelian groups, and the Heisenberg groups are the basic infinite family of extraspecial $p$-groups on which to test it.

What carries the argument

The proof is carried by four interacting tools. First, the ordering-value set $\Omega(B)$ of a sequence $B$ in $C_p^2$: for a fixed ordering the product's central coordinate is $q(I)$, and Lemma 3.3 gives $|\Omega(B)|\ge \min\{p,n-1\}$ for mixed zero-sum sequences of $n\ge 3$ nonzero terms, which lets the authors conclude that all central elements occur among the products of a length-$p^3$ mixed subsequence. Second, a maximal decomposition of the sequence into disjoint $p$-term zero-sum blocks in $C_p^2$, bounded by the classical result that every $4p-3$ terms contain such a block. Third, Lemma 3.12, a polynomial-method statement: if fewer than $3p$ vectors in $C_p^2$ avoid zero-sum subsequences of length $p$ and $2p$, then the subset sums of those lengths in a fixed order-$p$ subgroup have size at least $N-2p+2$; its proof uses a multilinear polynomial that vanishes on the Boolean cube and a degree count. Fourth, the final step isolates a fiber of height at least $p^3+p-1$ in one cyclic quotient, from which a $p^3$-term product-one subsequence is extracted by the classical prescribed-length theorem.

What would settle it

A direct check would settle the key lemma: for some odd prime $p$, search all mixed zero-sum sequences of $n=2p-1$ nonzero vectors in $C_p^2$ and test whether any has fewer than $p$ ordering values; one such example would invalidate Lemma 3.3 and with it the proof as written. For the theorem itself, a counterexample would be a sequence over $H_{p^3}$ of length $p^3+3p-3$ with no product-one subsequence of length $p^3$; exhaustive search for $p=3$ (a group of order 27, sequences of length 33) is already within computational reach.

Watch

Extended reading notes

Core claim

The central discovery is a parameter-free determination of the Gao constant for all odd primes $p$. Working with the canonical projection $\varphi: H_{p^3} \to H_{p^3}/Z \cong C_p^2$, the paper shows that a sequence of length $p^3+3p-3$ whose terms avoid a long block structure must concentrate $p^3+p-1$ terms in a single coset of the center, and such a concentration forces a product-one subsequence of length $p^3$. Together with the known small Davenport constant $d(H_{p^3})=3p-3$, this yields $E(H_{p^3}) = p^3+3p-3$.

Load-bearing premise

The proof hinges on a lower bound on the number of possible ordering-dependent central values that a zero-sum set of vectors in the plane over $\mathbb{F}_p$ can produce: it must be at least $\min\{p,n-1\}$ for $n\ge 3$ vectors not lying in one line through the origin. If this bound fails for some sequence of about $2p$ vectors, the step that makes every central element available would break down, and the rest of the argument would not go through.

Editorial extensions

If this is right

For every odd prime $p$, $E(H_{p^3}) = p^3+3p-3$, so the Zhuang-Gao equality holds for the whole family of exponent-$p$ Heisenberg groups, not only for $p=3$.
The threshold is optimal: since $d(H_{p^3})=3p-3$, the lower bound $E(G)\ge d(G)+|G|$ shows that a sequence of length $p^3+3p-4$ can avoid a product-one subsequence of length $p^3$.
Any sequence over $H_{p^3}$ of length at least $p^3+3p-3$ contains a product-one subsequence of length exactly $p^3$; by definition of the Gao constant, this is the least such length.
The result extends the classical abelian identity $E(G)=d(G)+|G|$ to an infinite family of nonabelian groups whose quotient by the center is elementary abelian.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural testable extension is to run the same block-decomposition plus ordering-value strategy for the larger extraspecial groups $H_{p^{2n+1}}$ of exponent $p$; the argument suggests the Gao constant may again equal the small Davenport constant plus the group order, but the ordering-value lower bound would need a higher-dimensional analogue.
The proof's reliance on ordering-value sets suggests that, for class-2 $p$-groups with abelian quotient, the Gao constant is governed by how many distinct central coordinates a mixed zero-sum sequence can realize; quantifying this for $C_p^n$ could yield a general lower-bound machinery.
The final fiber argument implies a structural dichotomy: any sequence just below the threshold either has a large central fiber or already contains a full-length product-one subsequence. Classifying the extremal sequences that fail to contain one could give inverse theorems for the Gao constant in this family.

Share X Bluesky LinkedIn Reddit HN

Formalized claims in Lean

Claim #1: The central discovery is a parameter-free determination of the Gao constant for all odd primes $p$. Working with the canonical projection $\varphi: H_{p^3} \to H_{p^3}/Z \cong C_p^2$, the paper shows that a sequence of length $p^3+3p-3$ whose terms avoid a long block structure must concentrate $p^3+p-1$ terms in a single coset of the center, and such a concentration forces a product-one subsequenc

/-- @claim 1 The central discovery is a parameter-free determination of the Gao constant for all odd primes $p$. Working with the canonical projection $\varphi: H_{p^3} \to H_{p^3}/Z \cong C_p^2$, the paper shows that a sequence of length $p^3+3p-3$ whose terms avoid a long block structure must concentrate $p^3+p-1$ terms in a single coset of the center, and such a concentration forces a product-one subsequenc -/
def central_claim : Prop :=

Signed reviews

No signed human review yet.

Request a human review

A listed scientist reviews the paper for a fee and the review publishes here regardless of verdict. See the reviewers or get listed.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper settles the Gao-Zhuang conjecture for the full family of exponent-p Heisenberg groups of order p^3. That is a real result, not an incremental step: only the p=3 case was known, and the general case required a new upper-bound argument. The authors determine E(H_{p^3}) = p^3 + 3p - 3, matching d(H_{p^3}) + |G|.

The proof is careful and, as far as I can tell, sound. The structure is standard for this area: a lower bound from Lemma 3.5 and Volkmann's Davenport constant, then a long upper-bound argument. The new additive ingredient is Lemma 3.12, a polynomial-method lemma that controls p-term and 2p-term sums in C_p^2; it is stated cleanly and the proof, aside from a typo or two, is correct. The final fiber argument, which forces a large fiber and then extracts a product-one subsequence of length p^3, is clever and checks out.

The soft spot is real but narrow. Claim 1, the linchpin of the whole proof, uses Lemma 3.3 quoted from Volkmann's recent preprint [17, Theorem 3.2] to argue that a certain set of ordering-dependent values fills F_p. Every later step relies on Claim 1. The lemma is not proved in this paper, and Volkmann's preprint is not yet independently verified. So the upper-bound proof is conditional on an external result. That is a dependency, not an observed error; the authors cite it explicitly and there is no sign of circularity. Still, if I were the editor I would ask the referee to verify Lemma 3.3 independently, or request that the authors include a proof as an appendix.

Minor issues: Section 2 duplicates the definition of pi and Pi_k, and Lemma 3.12 has a couple of typographical slips (e.g., 'eF(x)' where the polynomial is meant). These do not affect the mathematics.

Verdict: this deserves serious peer review. The result is important for zero-sum theory in nonabelian groups, the proof is mostly self-contained, and the cited dependencies are legitimate. I would send it to a referee who knows Volkmann's methods and ask them to check Lemma 3.3 carefully. If that lemma holds up, the paper should be accepted.

Referee Report

3 major / 4 minor

Summary. The paper proves the Zhuang-Gao conjecture for the Heisenberg group H_{p^3}=UT_3(F_p) for every odd prime p. Building on Volkmann's determination of the small Davenport constant d(H_{p^3})=3p-3, the authors show that the Gao constant satisfies E(H_{p^3})=d(H_{p^3})+|H_{p^3}|=p^3+3p-3. The proof uses the ordering-value method for mixed zero-sum sequences over C_p^2, a polynomial lower bound for p- and 2p-term subsums (Lemma 3.12), and a maximal block decomposition into p-term zero-sum subsequences in the quotient, leading to the main contradiction via a product-one subsequence of length p^3. The paper is organized into an introduction, a notation section, a section of auxiliary lemmas, and the main proof in Section 4.

Significance. If correct, the result gives the first infinite family of nonabelian p-groups of order p^3 for which the Gao constant is determined and matches the conjectured formula, resolving a question of Godara and Sarkar. It provides strong support for the Zhuang-Gao conjecture on a canonical family of extraspecial p-groups. The proof is an original derivation that combines Volkmann's ordering-value ideas with a self-contained polynomial argument and a delicate fiber-counting argument in the final step. The internal development is clear and checkable, and there is no circularity with the Gao constant being proved. However, the main theorem depends on two results from preprints—Lemma 3.3 (Volkmann) and Lemma 3.7 (the authors' own preprint)—and one auxiliary lemma (Lemma 3.12) has a small gap in its proof that is repairable.

major comments (3)

[Section 4, Claim 1] Claim 1 uses Lemma 3.3, quoted from [17, Theorem 3.2], to infer |Ω(φ(T0))|=p from |T0|≥2p+1. Since Claim 1 is used at every later step (the block decomposition into K, the exclusion of mixed length-p^3 projections, and the final fiber arguments), the proof of Theorem 1.1 is conditional on this lemma. Lemma 3.3 is a theorem from an arXiv preprint that is not proved here and is not available in a peer-reviewed venue; the authors should either provide a proof, or clearly state the result as an external dependency and ensure its status is verifiable.
[Section 4, first paragraph] Lemma 3.7, cited from the authors' preprint [14], is used to conclude v_z(S)≤3p-4 for every z∈Z, which is essential for the bound (11) on |S_Z|. This lemma is not proved in the paper. Please include a proof or a reference to a published result; otherwise the bound on |S_Z| is unsupported.
[Section 3, Lemma 3.12] In the proof of Lemma 3.12, the assertion that R is a nonempty subset of F_p\{0} is not justified by the argument given: if R were empty, then 0∉R, and the polynomial argument would still run but would yield only N≤2p-2, which is not a contradiction unless N≥2p-1. The proof should rule out the empty case explicitly (for example, by noting the degree comparison gives a contradiction when N≥2p-1) before defining P as a product over R. This is a repairable gap, but it occurs in a lemma that is load-bearing for the final fiber argument.

minor comments (4)

[Section 2] The definition of π(S) and Π_k(S) appears twice; the first occurrence (the paragraph before Eq. (2)) is a duplicate and should be removed.
[Section 3, Lemma 3.12] In the line 'Note that R is a nonempty subset of F_p\{0}, since otherwise, 0∈R...' the phrasing is imprecise; it should read 'if R contained 0, then...' or handle the empty case as described above.
[Section 4, after Eq. (19)] The statement 'L_K = L_{K\Z} = W' could be clarified, since L_K denotes the subsequence of L contained in K and W is a subsequence of S_{K\Z}; the equality is correct but the notation is introduced without explicit definition.
[References] Reference [14] is a preprint by the same authors; given that the proof relies on it, the authors should give a fuller statement of Lemma 3.7 and, ideally, a proof sketch.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: Theorem 1.1 is proved by direct contradiction using external lemmas; no fitted parameter is renamed a prediction.

full rationale

The proof of Theorem 1.1 derives E(H_{p^3}) ≥ p^3+3p−3 from the classical lower bound (Lemma 3.5) and Volkmann's d(H_{p^3})=3p−3 (Lemma 3.6), then proves the reverse inequality by assuming a product-one-free sequence of length p^3+3p−3 and deriving contradictions. The main internal steps—Claim 1, the block decomposition, Lemma 3.12's polynomial argument, and the final fiber-counting—do not presuppose E(G)=d(G)+|G|; they only use zero-sum facts over C_p^2 (Lemmas 3.4, 3.8, 3.9, 3.10) and the cited Omega-set lemma (Lemma 3.3). Lemma 3.7, the only self-citation used in a load-bearing way, is a general multiplicity lemma whose stated assumptions do not include the target Gao–Zhuang equality, so citing it does not reduce the theorem to its own conclusion. The paper contains no fitted parameters, no prediction-equivalent normalization, and no uniqueness assertion imported from the authors' earlier work. At most, the argument is conditional on the correctness of Volkmann's Lemma 3.3 and on the unpublished Lemma 3.7; that is an external-dependency/correctness risk, not circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No parameters are fitted and no new entities are introduced. The proof depends on recent theorems of Volkmann and on classical zero-sum results, plus a new polynomial-method lemma (Lemma 3.12) proved in the paper. The only self-citation is Lemma 3.7, from a preprint by members of the same group.

assumptions (7)

domain assumption Lemma 3.3 (from Volkmann): for odd p, every mixed zero-sum sequence of n>=3 nonzero terms from C_p^2 satisfies |Omega(B)|=|W(B)|>=min{p,n-1}.
Specialized recent result by Volkmann, not proved in this paper; used in Claim 1 (Section 4) to force pi(T)=Z.
domain assumption Lemma 3.6 (from Volkmann): d(H_{p^3})=3p-3 for every odd prime p.
Recent theorem by Volkmann; provides the lower bound in the main theorem and is used throughout Section 4.
standard math Lemma 3.4 (Gao): every sequence over an abelian group A of length at least kn+d(A) contains a zero-sum subsequence of length kn.
Classical zero-sum theorem, used for C_p and C_p^2 in the proof of Lemma 3.13 and the final central-fiber argument.
standard math Lemma 3.8 (Reiher): every sequence of length 4p-3 over C_p^2 has a zero-sum subsequence of length p.
Classical result, used repeatedly to obtain maximal block decompositions.
standard math Lemma 3.9 (Gao-Geroldinger): every sequence of length 3p-2 over C_p^2 has a zero-sum subsequence of length p or 2p.
Used to rule out the case t=p^2-1 in Section 4.
domain assumption Lemma 3.7 (from Qu-Gao-Li preprint): if v_z(S)>=d(G) for a central z, then S has a product-one subsequence of length |G|.
Self-citation to the authors' own preprint; a simple lemma but not proved in this paper.
domain assumption Lemmas 3.1 and 3.2 (from Volkmann): relation between q(I) and W(I), and the product formula for ordered terms of H_{p^3}.
Used in Claim 1 to link the product set of a zero-sum projection to Omega(phi(T)).

how reviews work

0 comments

Cite this review

Pith. "Pith review of The Gao-Zhuang conjecture for the Heisenberg group over $\mathbb{F}_p$." pith.science (2026). https://pith.science/paper/6OF7WOSE

@misc{pith2026260823319,
  author       = {Pith},
  title        = {Pith review of: The Gao-Zhuang conjecture for the Heisenberg group over $\mathbbF_p$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6OF7WOSE}},
  note         = {Machine review of arXiv:2608.23319}
}

abstract

Let $G$ be a finite nonabelian group. The small Davenport constant $\mathsf d(G)$ of $G$ is the largest integer $\ell$ such that there exists a product-one free sequence over $G$ of length $\ell$, while the Gao constant $E(G)$ of $G$ is the least integer $\ell$ such that every sequence over $G$ of length at least $\ell$ contains a product-one subsequence of length exactly $|G|$. A long-standing conjecture of Gao and Zhuang \cite{ZG2005} asserts that $E(G)=\mathsf d(G)+|G|$ for every finite nonabelian group $G$. Let $p$ be an odd prime and let $H_{p^3}=\operatorname{UT}_3(\mathbb F_p)$ be the finite Heisenberg group over $\mathbb F_p$. Godara and Sarkar proved the Gao-Zhuang equality for $H_{27}=\operatorname{UT}_3(\mathbb F_3)$ and asked whether the same equality holds for $H_{p^3}$ for every odd prime $p$. Recently, Volkmann proved that $\mathsf d(H_{p^3})=3p-3$. In this paper, we determine the Gao constant of $H_{p^3}$ and prove that $E(H_{p^3})=\mathsf d(H_{p^3})+|H_{p^3}|=p^3+3p-3$. 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 $p$.

Discussion (0). Continue with ORCID to comment.

Pith tools

Reviewed August 28, 2026 · model on record in the stance chip above.