Small Davenport constant of the rank-two extraspecial group

Establish that the small Davenport constant of the direct product H_2(F_3) × C_3^r, where H_2(F_3) is the extraspecial group of order 3^5 and exponent three, equals 2r + 10 for every integer r ≥ 0.

Background

The paper proves that the small Davenport constant of H_{27} × C_3r is 2r + 6 for every r ≥ 0, using a rank-two alternating form and a triangle-weighting argument. The authors explain that this method depends essentially on the two-dimensional quotient associated with the commutator form: in higher rank, the common orthogonal complement of two independent vectors can be nonzero, so isolated vertices in the nonorthogonality graph may contribute to the additive sum.

As a concrete higher-rank target, the paper identifies the direct product H_2(F_3) × C_3r, where H_2(F_3) is the extraspecial group of order 35 and exponent three, and proposes the formula 2r + 10 for all nonnegative integers r. The statement is explicitly left unproved in the paper.

References

A natural further target is

(H_2(\mathbb F_3)\times C_3r)=2r+10\qquad(r\geq0),

where $H_2(\mathbb F_3)$ is the extraspecial group of order $35$ and exponent three. This further statement is not proved here.

— The small Davenport constant of $H_{27}\times C_3^r$  (2609.19810 - Volkmann, 17 Sep 2026) in Section 5, “Structural formulation and scope”