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Strictness between small and large Davenport constants in non-abelian groups

Ascertain whether there exists a finite non-abelian group G with d(G) + 1 = D(G), or prove that d(G) + 1 < D(G) holds for every finite non-abelian group G.

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Background

For abelian groups, the relationship d(G) + 1 = D(G) is known. The authors note the absence of known non-abelian examples where equality holds and emphasize the question of whether strict inequality always occurs for non-abelian groups.

This problem concerns fundamental differences in factorization behavior between abelian and non-abelian settings and the gap between the small and large Davenport constants.

References

Unlike abelian groups, we are not aware of a finite non-abelian group with $\mathsf d (G) + 1 = \mathsf D (G)$, and so it is worthwhile to mention whether $\mathsf d (G) + 1 < \mathsf D (G)$ for all finite non-abelian group $G$.

— A classification of finite groups with small Davenport constant (2409.00363 - Oh, 31 Aug 2024) in Section 1 (Introduction)