The small Davenport constant of the Heisenberg group of order 125

Abstract

The small Davenport constant $\mathsf{d}(G)$ of a finite group $G$ is the maximal length of a product-one-free sequence over $G$. For the exponent-$p$ Heisenberg group $H_{p^3}$ of order $p^3$, Godara and Sarkar proved $\mathsf{d}(H_{27})=6$ and posed $\mathsf{d}(H_{p^3})=3p-3$ for every odd prime $p$, leaving $p\ge5$ open. We settle the first open case: $\mathsf{d}(H_{125})=12$. The lower bound is the explicit product-one-free sequence $x^4y^4v^4$. For the upper bound we record a product-one criterion that reduces the non-commutative problem to additive combinatorics over $\mathbb{F}_5^2$, and then reduce "every length-13 sequence has a product-one subsequence" to a single finite statement -- a spread bound on quotient multisets -- which we verify by an exhaustive, memory-flat search in C, its verdict independently reproduced by a second search with a different pruning strategy. Every auxiliary lemma is machine-checked. The argument is genuinely $p$-specific: we identify the exact step that fails for $p\ge7$ (a Chevalley-Warning shortcut whose forced block need not be wide), exhibit the obstructing multiset for $p=7$, and leave only $18\le\mathsf{d}(H_{343})\le24$. The techniques -- the Cauchy-Davenport theorem, Chevalley-Warning, and Olson's value of the Davenport constant of $C_p^2$ -- are standard; the contribution is their assembly against a new non-abelian target and the finite verification that closes it.