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A. Volkmann

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Preprint Aug 2026

The small Davenport constant of the Heisenberg group of order 343

For a finite group $G$, let $\mathsf{d}(G)$ denote the maximum length of a sequence having no nonempty subsequence whose terms can be ordered to have product one. For an odd prime $p$, let $H_{p^3}=\operatorname{UT}*3(\mathbb{F}*p)$. Godara and Sarkar proved $\mathsf{d}(H*{27})=6$ and conjectured $\mathsf{d}(H*{p^3})=3p-3$; in a recent preprint, White proved the next case $\mathsf{d}(H_{125})=12$ and left $18\leq\mathsf{d}(H_{343})\leq24$. We prove $\mathsf{d}(H_{343})=18$. We adopt White's product-one criterion and spread framework and develop a $p=7$-specific direction stratification. An explicit product-one-free sequence gives the lower bound. For the upper bound, we stratify a hypothetical product-one-free sequence of length $19$ by the number of central terms and by the occupied projective directions of its quotient multiset. Supports on at most two directions are excluded by a theoretical argument whose finite auxiliary statements are exhaustively checked; the three-direction case and the case of five central terms are settled by exact finite computations. The remaining thirty strata are encoded by a counterexample-guided SAT procedure. A separately implemented checker verifies all $9{,}920{,}815$ seed cuts and all $27{,}207$ learned cuts, and each final unsatisfiable instance is accompanied by a checked LRAT certificate. A separate implementation-level audit verifies the master encoding, the proof archives, and the lower-bound witness.

A. Volkmann · 1 citation
Preprint Aug 2026

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

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.

A. Volkmann · 1 citation