Godara–Sarkar conjecture for the small Davenport constant of Heisenberg groups

Godara and Sarkar conjectured that, for every odd prime $p$ , if $H_{p^3}=\operatorname{UT}_3(\mathbb{F}_p)$ is the Heisenberg group of order $p^3$ , then $\mathsf{d}(H_{p^3})=3p-3$ , where $\mathsf{d}(G)$ is the maximum length of a sequence over a finite group $G$ having no nonempty subsequence whose terms can be ordered to have product equal to the identity.

Sources & referencesView supporting material

Primary source

arXiv

Additional references

The small Davenport constant of the Heisenberg group of order 343 — arXiv — Andreas Volkmann

additive-and-nonabelian-combinatorics computer-assisted-proof finite-group-theory

4 days ago

Progress summary

Refreshed 2 days ago

Partially solved

Two new preprints claim the conjecture for the cases corresponding to primes five and seven, but the general statement remains unproved.

Godara and Sarkar conjectured that the small Davenport constant of the Heisenberg group equals $3p-3$ for every odd prime $p$ . They proved the smallest case, $p=3$ , and left $p\ge5$ open.

Known results

Godara and Sarkar: $\mathsf{d}(H_{27})=6$ .
A later paper proves the conjecture for a broad subclass of $2$ -generator, class-two $p$ -groups, but does not establish the full Heisenberg-group conjecture.

July–August 2026 claims for $p=5$ and $p=7$

White’s July preprint claims $\mathsf{d}(H_{125})=12$ , using an exhaustive computational verification of a finite spread bound. A separate August preprint claims $\mathsf{d}(H_{343})=18$ by excluding all $30$ strata for a hypothetical product-one-free sequence of length $19$ . Both are claims in arXiv preprints, not independently verified proofs in the retrieved sources.

Current status (as of August 2026): $p=3$ is settled, and preprints claim the conjectured values for $p=5$ and $p=7$ , but the claims remain unverified and the all-odd-prime conjecture is open.

Sources

No solutions have been posted yet.