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

1. Problem Statement and Context

The Core Problem:
The paper addresses the direct and inverse zero-sum problems for a specific class of non-abelian finite groups known as metacyclic groups. Specifically, it focuses on groups of the form:
$$G = C_n \rtimes_s C_2 = \langle x, y \mid x^2 = y^n = 1_G, yx = xys \rangle$$
where $s^2 \equiv 1 \pmod n$.

Key Definitions:

• Product-One Sequence: A finite sequence of elements from $G$ (order disregarded, repetitions allowed) is a product-one sequence if its elements can be ordered such that their product equals the identity element $1_G$.
• Gao's Constant $E(G)$: The smallest integer $\ell$ such that every sequence of length at least $\ell$ contains a product-one subsequence of length exactly $|G|$.
• Inverse Problem: Describing the structure of sequences of maximal length ($E(G)-1$) that do not contain a product-one subsequence of length $|G|$.

The Gap:
Previous work (specifically reference [3]) had settled the direct and inverse problems for almost all metacyclic groups of this form. However, a specific "exceptional" infinite family of cases remained unresolved. This family occurs when:

• $n = 3n_2$ where $n_2 > 1$ and $\gcd(n_2, 6) = 1$.
• $s \equiv -1 \pmod 3$ and $s \equiv 1 \pmod{n_2}$.
• In this case, $G \cong C_{n_2} \times D_6$ (a direct product of a cyclic group and the dihedral group of order 6).

The difficulty in this case arises because the structure of extremal sequences for $D_6$ (the $n_1=3$ case) is more complex than for $D_{2n_1}$ where $n_1 > 3$, causing previous inductive methods to fail.

2. Methodology

The authors employ a combination of combinatorial group theory, additive number theory, and structural analysis of sequences.

Key Tools and Techniques:

1. DeVos-Goddyn-Mohar Theorem (Lemma 3.1):
   This is the central technical innovation used to resolve the exceptional case. It is a generalization of the Cauchy-Davenport and Kneser theorems for abelian groups. The theorem provides a lower bound on the size of the set of $n$-subproducts ($\Pi_n(S)$) based on the stabilizer of the set of subproducts.

   • Application: The authors use this to analyze the projection of sequences onto the subgroup $\langle y^3 \rangle \cong C_{n_2}$. The theorem forces a dichotomy: either the set of subproducts covers the entire subgroup, or the sequence is highly concentrated in a specific coset of a stabilizer subgroup.

2. Decomposition and Projection:
   The group $G$ is decomposed as $C_{n_2} \times D_6$. The authors analyze sequences by projecting them onto the cyclic component $\langle y^3 \rangle$ and the dihedral component $\langle x, y^{n_2} \rangle$.

   • They utilize the known value $E(C_{n_2}) = 2n_2 - 1$ (Lemma 2.2) to extract product-one subsequences from the cyclic projection.

3. Inductive and Contradiction Arguments:

   • Proposition 4.1: The authors prove that any sequence of length $9n_2 - 1$ containing at least two elements from the coset $x\langle y \rangle$ must have a product-one subsequence of length $6n_2$.
   • Case Analysis: The proof of Proposition 4.1 involves a detailed case analysis based on the number of elements in the coset $x\langle y \rangle$ within a constructed sequence of "blocks" ($T_1, \dots, T_8$).
   • Swapping Subsequences: A crucial step involves swapping elements between subsequences to create new decompositions. By analyzing how the product of these blocks changes, they derive contradictions if the sequence is assumed to be product-one free.

4. Handling the Exceptional $D_6$ Structure:
   Unlike larger dihedral groups where extremal sequences have a uniform structure, the $D_6$ case allows for specific configurations (e.g., $1_G^{[5]} \cdot \tau \cdot \tau\alpha \cdot \tau\alpha^2$). The authors carefully handle these specific forms using the DeVos-Goddyn-Mohar theorem to show that even these "worst-case" configurations eventually yield a product-one subsequence if the length is sufficient.

3. Key Contributions and Results

The paper provides a complete solution to the Gao's constant and inverse problem for all metacyclic groups of the form $C_n \rtimes_s C_2$.

Main Theorem 1.1 (The Exceptional Case):
For the previously unresolved case $G = C_{3n_2} \rtimes_s C_2$ (where $n_2 > 1, \gcd(n_2, 6)=1, s \equiv -1 \pmod 3, s \equiv 1 \pmod{n_2}$):

1. Gao's Constant: $E(G) = 9n_2$.
2. Inverse Structure: A sequence $S$ of length $9n_2 - 1$ has no product-one subsequence of length $6n_2$ if and only if $S$ has the specific form:
   $$S = (\alpha^{t_1})^{[6n_2-1]} \cdot (\alpha^{t_2})^{[3n_2-1]} \cdot (\tau\alpha^{t_3})$$
   where $\alpha, \tau$ generate $G$, and $\gcd(t_1 - t_2, 3n_2) = 1$.

Main Theorem 1.2 (General Conclusion):
Combining their result with previous work, the authors establish the complete classification for all metacyclic groups $G = C_n \rtimes_s C_2$ (with $n \ge 3, s^2 \equiv 1 \pmod n, s \not\equiv 1 \pmod n$):

1. Gao's Constant: $E(G) = 3n$.
2. Inverse Structure:
   • If $n=3$: The extremal sequences are either $1_G^{[5]} \cdot \tau \cdot \tau\alpha \cdot \tau\alpha^2$ or of the form $(\alpha^{k_1})^{[5]} \cdot (\alpha^{k_2})^{[2]} \cdot \tau\alpha^{k_3}$.
   • If $n \ge 4$: The extremal sequences are of the form $(\alpha^{t_1})^{[2n-1]} \cdot (\alpha^{t_2})^{[n-1]} \cdot \tau\alpha^{t_3}$ with $\gcd(t_1 - t_2, n) = 1$.

4. Significance

• Completeness of Theory: This paper closes the final gap in the study of Gao's constants for metacyclic groups of the form $C_n \rtimes C_2$. It confirms that the formula $E(G) = 3n$ holds universally for this class, regardless of the specific parameters of the semidirect product.
• Methodological Advancement: The successful application of the DeVos-Goddyn-Mohar Theorem to a non-abelian zero-sum problem demonstrates the power of additive combinatorics tools in analyzing non-abelian group structures. It provides a new template for handling cases where standard inductive arguments fail due to the complexity of small subgroups (like $D_6$).
• Structural Insight: The paper provides a precise characterization of the "extremal" sequences (those just below the threshold). This is vital for understanding the boundary conditions of zero-sum phenomena in non-abelian settings.
• Future Directions: The authors note in the concluding remarks that while split metacyclic groups are now fully understood, the non-split metacyclic groups (where the extension does not split) with cyclic subgroups of index 2 remain an open area for future research.

In summary, this work represents a definitive resolution to a long-standing open problem in combinatorial number theory, unifying the theory of Gao's constants for a broad class of non-abelian groups through sophisticated structural analysis and the application of deep additive number theory results.