Davenport constant of $G=C_p^{d-1}\times C_{pq}$ for rank $d\ge 3$: settling Conjecture 1 of Biswas–Mazumdar and deciding $D(G)=D^*(G)$ across the family

For a finite abelian group $G$ written as $C_{n_1}\oplus\cdots\oplus C_{n_d}$ with $n_1\mid\cdots\mid n_d$, the Davenport constant $D(G)$ is the least $k$ such that every sequence of $k$ elements of $G$ has a nonempty zero-sum subsequence, and $D^*(G)=1+\sum_i(n_i-1)$ is its standard lower bound. Whether $D(G)=D^*(G)$ is a classical open question, undecided already for groups of rank 3. Biswas and Mazumdar (arXiv:2402.09999; Mathematical Notes 118 (2025) 23–34) studied $G=C_p^{d-1}\times C_{pq}$ of rank $d\ge 3$: for coprime $(p,q)$ they reduced the equality $D(G)=D^*(G)$ to an explicit combinatorial criterion on structured sequences over $C_p^d\times C_q$ (their Conjecture 1), verified only for $p=3,d=3$; for non-coprime $(p,q)$ they proved the equality only for sufficiently large primes $p$, via the r-wise Davenport constant $D_r(G)$ (the least $k$ such that every $G$-sequence of length at least $k$ contains $r$ pairwise disjoint zero-sum subsequences). The open problem stated here is to settle that criterion and to decide the equality for the entire family: prove or refute Conjecture 1, decide whether $D(G)=D^*(G)$ holds unconditionally for all $(p,q)=1$, and determine whether the large-$p$ hypothesis can be removed for $(p,q)\ne 1$. The problem remains open as of 2026.

Background

A sequence over a finite abelian group $G$ (written additively) is a finite list of elements of $G$ in which repetition is allowed; it has a zero-sum subsequence if some nonempty sub-list sums to the identity. The Davenport constant $D(G)$ is the least integer $k$ such that every $G$-sequence of length at least $k$ contains a nonempty zero-sum subsequence. It governs, for example, the arithmetic of ideal class groups of algebraic number fields (finite abelian groups measuring the failure of unique factorization of ideals) and, more generally, the theory of non-unique factorizations in Krull monoids (commutative monoids, such as the multiplicative monoid of nonzero elements of a ring of algebraic integers, whose elements can admit essentially different factorizations). Every finite abelian group decomposes as $G\cong C_{n_1}\oplus\cdots\oplus C_{n_d}$ with $n_1\mid\cdots\mid n_d$ (here $C_n$ is the cyclic group of order $n$); $d$ is the rank and $n_d$ the exponent of $G$. Taking $n_i-1$ copies of a generator of each cyclic factor $C_{n_i}$ and concatenating them yields a zero-sum-free sequence of length $\sum_{i=1}^{d}(n_i-1)$, whence

$$D(G)\;\ge\;D^*(G):=1+\sum_{i=1}^{d}(n_i-1),$$

and a long-standing conjecture of Olson asserts $D(G)=D^*(G)$ for all finite abelian groups. Olson proved equality for rank at most 2 and for all finite abelian $p$-groups; Geroldinger and Schneider constructed infinitely many groups of ranks 4 and 5 with $D(G)>D^*(G)$, so the conjecture fails in general, and whether any rank-3 group satisfies $D(G)>D^*(G)$ remains a famous open problem.

Biswas and Mazumdar attack this circle for the family $G=C_p^{d-1}\times C_{pq}$ of rank $d\ge 3$, where $p$ is prime and $q\in\mathbb{N}$; note $D^*(G)=(d-1)(p-1)+pq$. The one previously known coprime case is the result of Bhowmik and Schlage-Puchta that $D(C_3\oplus C_3\oplus C_{3s})=D^*(G)$ for every $s\ge 1$, i.e. the case $p=3$, $d=3$. For $(p,q)=1$ one has $C_{pq}\cong C_p\oplus C_q$, hence $G\cong C_p^d\times C_q$, and they reduce the equality $D(G)=D^*(G)$ to an explicit criterion on sequences of length $m=p(q+d-1)-(d-1)=D^*(G)$ over $C_p^d\times C_q$ (Conjecture 1 below). For $(p,q)\ne 1$ they instead use the r-wise Davenport constant $D_r(G)$ — introduced by Halter-Koch — which is the least $k$ such that every $G$-sequence of length at least $k$ contains $r$ pairwise disjoint zero-sum subsequences; building on the Freeze–Schmid theory of $D_r$ and on their own exact formulas for $D_r$ of $p$-groups whose largest invariant dominates the others, they prove $D(G)=D^*(G)$ for this case whenever the prime $p$ is sufficiently large. Exact values of $D_r$ are known for groups of rank at most 2 and for rank-3 groups $C_2\oplus C_{n_2}\oplus C_{n_3}$ with $2\mid n_2\mid n_3$ (Girard–Schmid), but no general formula for $D_r((C_p)^d)$ with $d\ge 3$ is known, and this gap is exactly what blocks the small-prime non-coprime case.

Problem Statement

The problem is the exact determination of the Davenport constant on the family $G=C_p^{d-1}\times C_{pq}$ of rank $d\ge 3$ ($p$ prime, $q\in\mathbb{N}$), i.e. deciding for which parameters $D(G)=D^*(G)=(d-1)(p-1)+pq$ holds. It has two linked parts.

(a) Coprime case. Since $(p,q)=1$ gives $G\cong C_p^d\times C_q$, Biswas and Mazumdar reduce the equality to the following self-contained criterion, stated there as Conjecture 1 and verified only for $p=3$, $d=3$:

Conjecture 1. Let $p$ be prime, $q\in\mathbb{N}$ with $(p,q)=1$, $d\ge 3$, and $m=p(q+d-1)-(d-1)$. Let $x_1\cdots x_m$ be an arbitrary sequence over $C_p^d$, and let the sequence $y_1\cdots y_m$ over $C_q$ have the block form $y=1^{r_1}\,2^{r_2}\,\cdots\,(q-1)^{r_{q-1}}\,0^{m-r}$ with $r=\sum_{i=1}^{q-1}r_i$ (elements of $C_q$ are written as the residues $0,1,\dots,q-1$, and $a^t$ denotes $t$ consecutive occurrences of the element $a$). If

$$r\in[\,pq+1,\;p(q+d-1)-(d-1)\,]\quad\text{and}\quad\sum_{i=1}^{q-1} i\,r_i\equiv 0 \pmod q,$$

then the sequence $S=(x_1,y_1)\cdots(x_m,y_m)$ over $C_p^d\times C_q$ has a nonempty zero-sum subsequence.

Prove or refute this conjecture. A proof, combined with Biswas and Mazumdar's Theorem 2, yields the unconditional equality $D(C_p^{d-1}\times C_{pq})=D^*(G)$ for every prime $p$ and every $q$ with $(p,q)=1$, $d\ge 3$; a refutation (an explicit structured sequence with no nonempty zero-sum subsequence), if the failure could be realized at $d=3$ by a zero-sum-free sequence of length $D^*(G)$, would produce the first rank-3 group with $D(G)>D^*(G)$.

(b) Non-coprime case. For $(p,q)\ne 1$ Biswas and Mazumdar prove $D(G)=D^*(G)$ only for sufficiently large primes $p$ (Theorem 7), via exact formulas for the r-wise Davenport constant $D_r$ of $p$-groups whose largest invariant dominates. Determine for which primes $p$ the equality $D(C_p^{d-1}\times C_{pq})=D^*(G)$ holds for all $q$ with $(p,q)\ne 1$ and all $d\ge 3$; a prime is exceptional exactly when $C_p^{d-1}\times C_{pq}$ admits a zero-sum-free sequence of length $D^*(G)$.

Part (a) and part (b) are the two coprime/non-coprime halves of one objective — deciding the Olson equality $D(G)=D^*(G)$ across the whole family $C_p^{d-1}\times C_{pq}$.

References

1. Anamitro Biswas, Eshita Mazumdar, Davenport constant for finite abelian groups with higher rank, arXiv:2402.09999 (2024); published as Mathematical Notes 118 (2025), no. 1–2, 23–34, DOI 10.1134/S0001434624605483, https://arxiv.org/abs/2402.09999
2. John E. Olson, A combinatorial problem on finite abelian groups, I, Journal of Number Theory 1 (1969) 8–10, DOI 10.1016/0022-314X(69)90021-3, https://doi.org/10.1016/0022-314X(69)90021-3
3. John E. Olson, A combinatorial problem on finite abelian groups, II, Journal of Number Theory 1 (1969) 195–199, DOI 10.1016/0022-314X(69)90037-7, https://doi.org/10.1016/0022-314X(69)90037-7
4. Alfred Geroldinger, Rudolf Schneider, On Davenport's constant, Journal of Combinatorial Theory, Series A 61 (1992) 147–152, DOI 10.1016/0097-3165(92)90061-X, https://doi.org/10.1016/0097-3165(92)90061-X
5. Gautami Bhowmik, Jan-Christoph Schlage-Puchta, Davenport's constant for groups of the form $\mathbb{Z}_3\oplus\mathbb{Z}_3\oplus\mathbb{Z}_{3d}$, in: Additive Combinatorics, CRM Proceedings and Lecture Notes 43, American Mathematical Society (2007) 307–326, DOI 10.1090/crmp/043/17, https://doi.org/10.1090/crmp/043/17
6. Benjamin Girard, Wolfgang A. Schmid, Direct zero-sum problems for certain groups of rank three, Journal of Number Theory 197 (2019) 297–316, DOI 10.1016/j.jnt.2018.08.016, arXiv:1806.07636, https://arxiv.org/abs/1806.07636
7. Franz Halter-Koch, A generalization of Davenport's constant and its arithmetical applications, Colloquium Mathematicum 63 (1992) 203–210, DOI 10.4064/cm-63-2-203-210, https://doi.org/10.4064/cm-63-2-203-210
8. Michael Freeze, Wolfgang A. Schmid, Remarks on a generalization of the Davenport constant, Discrete Mathematics 310 (2010) 3373–3389, DOI 10.1016/j.disc.2010.07.028, https://doi.org/10.1016/j.disc.2010.07.028
9. Luz E. Marchan, Oscar Ordaz, Irene Santos, Wolfgang A. Schmid, Multi-wise and constrained fully weighted Davenport constants and interactions with coding theory, Journal of Combinatorial Theory, Series A 135 (2015) 237–267, DOI 10.1016/j.jcta.2015.05.004, arXiv:1407.1966, https://arxiv.org/abs/1407.1966
10. Maciej Zakarczemny, Note on the Davenport's constant for finite abelian groups with rank three, arXiv:1910.10984 (2019), https://arxiv.org/abs/1910.10984
11. Alexander Towell, Davenport Constants for Rank-3 Finite Abelian Groups, Zenodo record 19211134 (2026), DOI 10.5281/zenodo.19211134, https://zenodo.org/records/19211134
12. Jingxuan Shi, Yushuang Fan, Yuanlin Li, Supremum of the union of sets of lengths containing k in Krull monoids, Communications in Algebra (2026), DOI 10.1080/00927872.2026.2647178, https://doi.org/10.1080/00927872.2026.2647178