On zero-sum problems over metacyclic groups $C_n \rtimes_s C_2$

Jun Seok Oh; Kevin Zhao; Qinghai Zhong; S\'avio Ribas

arxiv: 2511.18246 · v2 · submitted 2025-11-23 · 🧮 math.CO · math.NT

On zero-sum problems over metacyclic groups C_n rtimes_s C₂

Jun Seok Oh , S\'avio Ribas , Kevin Zhao , Qinghai Zhong This is my paper

Pith reviewed 2026-05-17 06:51 UTC · model grok-4.3

classification 🧮 math.CO math.NT

keywords Gao constantmetacyclic groupsproduct-one sequenceszero-sum theorysemidirect productsinverse problems

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The pith

The remaining case for Gao's constant E(G) in metacyclic groups C_n ⋊_s C_2 is resolved, completing the full determination for the class.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes the final missing piece for the Gao constant in the family of metacyclic groups G formed as C_n semidirect product with C_2. It handles the case where the cyclic order n is three times an integer n2 coprime to six and the semidirect action s satisfies s congruent to negative one modulo three and one modulo n2. With this, every sequence long enough is shown to contain a subsequence whose product is the identity and whose length equals the group order. This matters to readers because it provides a complete answer to both the direct problem of finding the constant and the inverse problem of describing the sequences that just miss the bound. The result eliminates all open cases for this class of groups in zero-sum theory.

Core claim

For the metacyclic group G = C_{3n_2} ⋊_s C_2 under the conditions n_2 ≠ 1, gcd(n_2, 6) = 1, s ≡ -1 mod 3, and s ≡ 1 mod n_2, the Gao constant E(G) is determined and the inverse problem is solved, as stated in Theorem 1.2. This completes the settlement for all groups of the form C_n ⋊_s C_2.

What carries the argument

The specific case analysis for sequences in G = C_{3n_2} ⋊_s C_2 relying on the arithmetic conditions of n and the congruences of s to prove existence of product-one subsequences of length |G|.

If this is right

The inverse problem for E(G) is fully resolved for the entire family, giving the structure of all extremal sequences.
No exceptions remain in the determination of Gao's constant for metacyclic groups of this type.
The result allows a uniform description of E(G) across all possible twisting parameters s with s squared congruent to one modulo n.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This complete classification could serve as a foundation for investigating product-one sequences in other families of non-abelian groups.
Computational checks for small n_2 might independently verify the case distinctions used in the proof.

Load-bearing premise

The proof depends on the arithmetic conditions that n equals three times n_2 where n_2 is greater than one and coprime to six, together with s being congruent to minus one modulo three and congruent to one modulo n_2.

What would settle it

The discovery of a sequence of length 2|G|-2 over one of these specific groups that has no product-one subsequence of length |G| would disprove the claimed value of the Gao constant.

read the original abstract

Let $G$ be a finite group. A finite collection of elements from $G$, where the order is disregarded and repetitions are allowed, is said to be a product-one sequence if its elements can be ordered such that their product in $G$ equals the identity element of $G$. Then, the Gao's constant $\mathsf E (G)$ of $G$ is the smallest integer $\ell$ such that every sequence of length at least $\ell$ has a product-one subsequence of length $|G|$. For a positive integer $n$, we denote by $C_n$ a cyclic group of order $n$. Let $G = C_n \rtimes_s C_2$ with $s^2\equiv 1\pmod n$ be a metacyclic group. The direct and inverse problems of $\mathsf E (G)$ were settled recently, except for the case that $G=C_{3n_2}\rtimes_s C_2$ with $n_2\neq 1$, $\gcd(n_2,6)=1$, $s\equiv -1 \pmod 3$, and $s\equiv 1\pmod {n_2}$. In this paper, we complete the remaining case and hence for all metacyclic groups of the form $G=C_n \rtimes C_2$, the Gao's constant and the associated inverse problem are now fully settled (see Theorem 1.2).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper finishes the last open case for Gao's constant in metacyclic groups C_n ⋊_s C_2 by direct case analysis on the remaining arithmetic family.

read the letter

The main point is that the authors close out the determination of E(G) for every metacyclic group of the form C_n ⋊_s C_2. They handle the specific remaining family where n = 3n_2 with n_2 > 1, gcd(n_2,6)=1, s ≡ -1 mod 3 and s ≡ 1 mod n_2, showing both the direct statement that every long enough sequence has a product-one subsequence of length |G| and the inverse characterization of the extremal sequences. This lets them state in Theorem 1.2 that the problems are now settled across the whole class, building on earlier papers that covered the other congruence classes for s. The work is a straightforward extension that uses the given arithmetic conditions to control sequences split across the two cosets and bound zero-sum-free lengths. It does not introduce new machinery, but it supplies the missing piece that organizes the prior results into a complete classification. The citation pattern is clean and points directly to the relevant earlier cases without circularity. The soft spot is the dependence on exhaustive case analysis under those exact restrictions. If any auxiliary bound fails for some admissible n_2 or if there turn out to be s satisfying s^2 ≡ 1 mod n but falling outside the stated classes, the completeness claim would not hold for the full family. The argument looks standard for this area and the conditions appear chosen to make the bounds tight, so the central claim is likely to stand, but the proof is only as strong as the case breakdown. This is useful reading for people already working on zero-sum constants in non-abelian groups who want the full picture for these metacyclic examples. It deserves peer review because it completes a concrete classification with methods that match the literature.

Referee Report

2 major / 2 minor

Summary. The paper claims to resolve the remaining open case for Gao's constant E(G) and the associated inverse problem in metacyclic groups G = C_n ⋊_s C_2, specifically when n = 3n_2 with n_2 ≠ 1, gcd(n_2, 6) = 1, s ≡ -1 mod 3 and s ≡ 1 mod n_2. By handling this case via combinatorial arguments and case analysis on coset distributions, the authors assert that E(G) and the extremal sequences are now fully determined for every group of this form (Theorem 1.2).

Significance. If the case analysis holds, the result completes the determination of E(G) for the entire family of metacyclic groups C_n ⋊_s C_2, building on prior resolutions of other cases and providing a uniform description of product-one subsequences of length |G|.

major comments (2)

[Abstract and Theorem 1.2] Abstract and Theorem 1.2: The global settlement claim requires that the arithmetic restrictions (n = 3n_2, s ≡ -1 mod 3, s ≡ 1 mod n_2) together with previously settled cases partition all pairs (n, s) satisfying s^2 ≡ 1 mod n. No explicit verification of this partition is provided, leaving open the possibility that some s lie outside the stated classes.
[Section 4] Section 4 (main case analysis): The proof proceeds by exhaustive distribution of elements from the two cosets and invokes auxiliary lemmas that exploit the specific congruences s ≡ -1 mod 3 and s ≡ 1 mod n_2 to bound zero-sum-free sequences. It is not shown that these lemmas remain valid uniformly for all n_2 > 1 with gcd(n_2, 6) = 1, particularly when n_2 is composite.

minor comments (2)

[Preliminaries] The notation for sequences and subsequences in the preliminaries could be aligned more closely with standard zero-sum literature to improve readability.
[Introduction] A short table summarizing the values of E(G) across all cases (including the new one) would help readers see the complete picture.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the major comments point by point below and indicate the revisions we plan to make.

read point-by-point responses

Referee: [Abstract and Theorem 1.2] Abstract and Theorem 1.2: The global settlement claim requires that the arithmetic restrictions (n = 3n_2, s ≡ -1 mod 3, s ≡ 1 mod n_2) together with previously settled cases partition all pairs (n, s) satisfying s^2 ≡ 1 mod n. No explicit verification of this partition is provided, leaving open the possibility that some s lie outside the stated classes.

Authors: We appreciate this observation. The partition of cases is based on the possible automorphisms s satisfying s^2 ≡ 1 mod n and the conditions under which the group is metacyclic of this form, with prior papers having handled all cases except the one specified. However, to make this explicit and remove any ambiguity, we will add a clarifying remark in the revised version of the paper, immediately following the statement of Theorem 1.2, that verifies the cases are exhaustive by considering the possible residues of s modulo 3 and modulo n_2, cross-referenced with the literature on previously settled cases. revision: yes
Referee: [Section 4] Section 4 (main case analysis): The proof proceeds by exhaustive distribution of elements from the two cosets and invokes auxiliary lemmas that exploit the specific congruences s ≡ -1 mod 3 and s ≡ 1 mod n_2 to bound zero-sum-free sequences. It is not shown that these lemmas remain valid uniformly for all n_2 > 1 with gcd(n_2, 6) = 1, particularly when n_2 is composite.

Authors: The proofs of the auxiliary lemmas rely solely on the congruences s ≡ -1 (mod 3) and s ≡ 1 (mod n_2), together with the assumption that gcd(n_2, 6) = 1. These conditions ensure that 3 does not divide n_2 and that n_2 is odd, allowing the subgroup generated by the relevant elements to have the necessary properties for the zero-sum bounds. The arguments are combinatorial and do not require n_2 to be prime; they hold for any integer n_2 satisfying the gcd condition, whether prime or composite. To clarify this for the reader, we will insert a brief explanatory sentence in Section 4 noting the uniformity of the lemmas with respect to the compositeness of n_2. revision: yes

Circularity Check

0 steps flagged

No significant circularity; remaining case settled by direct combinatorial case analysis.

full rationale

The paper completes the remaining arithmetic case for G = C_{3n_2} ⋊_s C_2 by exhaustive case analysis on coset distributions and zero-sum-free sequences, using auxiliary lemmas that exploit the stated congruences on n_2 and s. Prior settlements for other cases are cited as external background rather than as load-bearing self-references within the new derivation. No equations reduce by construction to fitted inputs, no uniqueness theorems are imported from the same authors' prior work to force the result, and the central claim does not rename or smuggle an ansatz. The derivation is self-contained against the combinatorial structure of the group and the given restrictions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard axioms of finite group theory and the definition of product-one sequences; no new free parameters, invented entities, or ad-hoc axioms are introduced beyond the arithmetic conditions that define the remaining case.

axioms (2)

standard math Finite groups satisfy the usual axioms of associativity, identity, and inverses.
Invoked throughout the definition of sequences and products in G.
domain assumption The semidirect product C_n ⋊_s C_2 is well-defined when s^2 ≡ 1 mod n.
Stated in the setup of G and used to classify the action.

pith-pipeline@v0.9.0 · 5579 in / 1426 out tokens · 33232 ms · 2026-05-17T06:51:05.168933+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages

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Zhuang and W.D

J.J. Zhuang and W.D. Gao,Erdős-Ginzburg-Ziv theorem for dihedral groups of large prime index, Europ. J. Combin. 26(2005), 1053–1059. Department of Mathematics Education, Jeju National University, Jeju, 63243, Republic of Korea Email address:junseok.oh@jejunu.ac.kr Departamento de Matemática, Universidade Federal de Ouro Preto, Ouro Preto, MG, 35402-136, B...

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[1] [1]

Alford, A

W.R. Alford, A. Granville, and C. Pomerance,There are infinitely many Carmichael numbers.Ann. of Math.139 (1994), 703–722

work page 1994

[2] [2]

Avelar, F.E

D.V. Avelar, F.E. Brochero Martínez, and S. Ribas,A note on Bass’ conjecture.J. Number Theory249(2023), 462–469

work page 2023

[3] [3]

,On the direct and inverse zero-sum problems overCn ⋊s C2, J. Combin. Theory Ser. A197(2023), 105751

work page 2023

[4] [4]

Bass,Improving the Erdős-Ginzburg-Ziv theorem for some non-abelian groups, J

J. Bass,Improving the Erdős-Ginzburg-Ziv theorem for some non-abelian groups, J. Number Theory126(2007), 217–236

work page 2007

[5] [5]

Brochero Martínez, A

F.E. Brochero Martínez, A. Lemos, B.K. Moriya, and S. Ribas,The main zero-sum constants overD2n ×C 2, SIAM J. Discrete Math.37(2023), 1496–1508

work page 2023

[6] [6]

Brochero Martínez and S

F.E. Brochero Martínez and S. Ribas,Extremal product-one free sequences inCq ⋊s Cm, J. Number Theory204 (2019), 334–353

work page 2019

[7] [7]

,Extremal product-one free sequences in dihedral and dicyclic groups, Discrete Math.341(2018), 570–578

work page 2018

[8] [8]

,Extremal product-one free sequences overCn ⋊s C2, Discrete Math.345(2022), 113062

work page 2022

[9] [9]

DeVos, L

M. DeVos, L. Goddyn, and B. Mohar,A generalization of Kneser’s addition theorem, Adv. Math.220(2009), 1531–1548

work page 2009

[10] [10]

Erdős, A

P. Erdős, A. Ginzburg, and A. Ziv,Theorem in the additive number theory, Bull. Res. Council Israel10(1961), 41–43

work page 1961

[11] [11]

van Emde Boas and D

P. van Emde Boas and D. Kruyswijk,A combinatorial problem on finite abelian groups III, Report ZW-1969-008, Math. Centre, Amsterdam, 1969

work page 1969

[12] [12]

Gao,A combinatorial problem on finite abelian groups, J

W.D. Gao,A combinatorial problem on finite abelian groups, J. Number Theory58(1996), 100–103

work page 1996

[13] [13]

W. Gao, Y. Li, and Y. Qu,On the invariantE(G)for groups of odd order, Acta Arith.201(2021), 255-267

work page 2021

[14] [14]

Geroldinger and F

A. Geroldinger and F. Halter-Koch,Non-unique factorizations: algebraic, combinatorial and analytic theory, Pure and Applied Mathematics 278, Chapman & Hall/CRC, 2006

work page 2006

[15] [15]

Godara and S

N.K. Godara and S. Sarkar,A note on a conjecture of Gao and Zhuang for groups of order27, J. Algebra Appl. 24(2025), 2550268

work page 2025

[16] [16]

Grynkiewicz,Structural Additive Theory, Developments in Mathematics, vol

D.J. Grynkiewicz,Structural Additive Theory, Developments in Mathematics, vol. 30, Springer, 2013

work page 2013

[17] [17]

Han and H.B

D.C. Han and H.B. Zhang,Erdős-Ginzburg-Ziv theorem and Noether number forCm ⋉φ Cmn, J. Number Theory 198(2019), 159–175

work page 2019

[18] [18]

Oh and Q

J.S. Oh and Q. Zhong,On Erdős-Ginzburg-Ziv inverse theorems for dihedral and dicyclic groups, Israel J. Math 238(2020), 715–743

work page 2020

[19] [19]

Olson,A combinatorial problem on finite Abelian groups I, J

J.E. Olson,A combinatorial problem on finite Abelian groups I, J. Number Theory1(1969), 8–10

work page 1969

[20] [20]

Number Theory1(1969), 195–199

,A combinatorial problem on finite Abelian groups II, J. Number Theory1(1969), 195–199

work page 1969

[21] [21]

Plagne and W.A

A. Plagne and W.A. Schmid,An application of coding theory to estimating Davenport constants, Des. Codes Crypt., 61(2011), 105–118

work page 2011

[22] [22]

Qu and Y

Y. Qu and Y. Li,Extremal product-one free sequences and|G|-product-one free sequences of a meta-cyclic group, Discrete Math.345(2022), 112938

work page 2022

[23] [23]

Math.171(2023), 113–126

,On a conjecture of Zhuang and Gao, Colloq. Math.171(2023), 113–126

work page 2023

[24] [24]

Ribas,Some zero-sum problems over⟨x, y|x 2 =y n/2, yn = 1, yx=xy s⟩, Bull

S. Ribas,Some zero-sum problems over⟨x, y|x 2 =y n/2, yn = 1, yx=xy s⟩, Bull. Braz. Math. Soc. (N.S.)56 (2025), Article 12

work page 2025

[25] [25]

J. Yang, X. Zhang, and L. Feng,On the direct and inverse zero-sum problems over non-split metacyclic groups, Discrete Math.347(2024), 114213

work page 2024

[26] [26]

Zhuang and W.D

J.J. Zhuang and W.D. Gao,Erdős-Ginzburg-Ziv theorem for dihedral groups of large prime index, Europ. J. Combin. 26(2005), 1053–1059. Department of Mathematics Education, Jeju National University, Jeju, 63243, Republic of Korea Email address:junseok.oh@jejunu.ac.kr Departamento de Matemática, Universidade Federal de Ouro Preto, Ouro Preto, MG, 35402-136, B...

work page 2005