A Uniform Proof for the Small Davenport Constant of the Exponent-$p$ Heisenberg Group

Abstract

Let $p$ be an odd prime and let $H_{p^3}=\operatorname{UT}_3(\mathbb{F}_p)$ be the Heisenberg group of order $p^3$ and exponent $p$. We prove $\mathsf{d}(H_{p^3})=3p-3$. The main ingredient of the proof is an order-value growth theorem. If $B$ is a noncollinear zero-sum sequence of $n$ nonzero vectors in $\mathbb{F}_p^2$, then the alternating areas obtained by ordering $B$ assume at least $\min(p,n-1)$ distinct values. Its proof is a short contraction induction: contract a suitable independent pair, replace the contracted vector in both orders, and apply Cauchy--Davenport. A polynomial relative-subsum theorem and a sharp representation-rigidity lemma then turn this local growth into a uniform spread bound. Combined with the standard product-one criterion for $H_{p^3}$, the spread bound yields the upper bound; the usual sequence $x^{p-1}y^{p-1}v^{p-1}$ gives the lower bound.

Disclosure

“l consistency checks during discovery; no computation is part of the logical argument. The proof itself uses only Cauchy–Davenport, elementary linear algebra, and the Boolean-cube degree argument given above. AI-assisted tools disclosure. AI-assisted tools were used during exploration, drafting, and adversarial consistency checking. The mathematical argument presented here is self-contained; no output of an AI system is invoked as an external authority. References [1] N. K. Godara an”

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Classification

Drafting limited passages

Multiplier

5

Verified

2026-08-22T18:30:51.911497Z

Structural counts