Gao's Constant in Finite Group Zero-Sum Theory

• Gao’s Constant is a combinatorial invariant defining the minimal sequence length required in a finite group to ensure a product-one (zero-sum) subsequence of length |G|.
• It establishes key bounds and formulas, such as the abelian case E(G)=d(G)+|G|-1, with extensions to non-abelian, metacyclic, and weighted settings.
• Research on Gao’s Constant drives insights in extremal combinatorics, factorization, and invariant theory, with proven results in dihedral, metacyclic, and nilpotent groups.

Gao’s constant is a central combinatorial invariant in modern zero-sum theory, describing the minimal sequence length in a finite group $G$ required to guarantee the existence of a product-one—or in the abelian context, zero-sum—subsequence of length %%%%1%%%%. Originally formulated for abelian groups, the notion generalizes to non-abelian and weighted settings and sits at the foundation of, and often determines extremal boundaries for, numerous direct and inverse problems in zero-sum theory, factorization theory, and finite group combinatorics.

1. Definition and Basic Properties

Let $G$ be a finite group. A "sequence" $S$ over $G$ is a finite multiset of $G$, often denoted $S = g_1 \cdot g_2 \cdots g_\ell$. The central zero-sum invariants are:

• Small Davenport constant, $d(G)$: the maximal length of a product-one-free sequence (no non-empty subsequence has product equal to the identity).
• Gao’s constant ($\mathsf E(G)$): the smallest integer $\ell$ such that any sequence of length at least $\ell$ over $G$ admits a product-one subsequence of length exactly $|G|$ (Zakarczemny, 2019, Avelar et al., 2022, Ribas, 6 Jan 2025, Oh et al., 23 Nov 2025, Martínez et al., 2021).

In abelian groups, products are replaced with sums, and Gao’s constant specializes as the minimal $t$ such that any sequence of $t$ elements contains a zero-sum subsequence of length $|G|$.

A key result for abelian groups is the classical formula: $\mathsf E(G) = d(G) + |G| - 1$ where $d(G)$ is the Davenport constant. For non-abelian groups, the Zhuang–Gao conjecture predicts: $\mathsf E(G) = d(G) + |G|$ This formula is verified in various non-abelian families (e.g., dihedral, metacyclic, dicyclic groups, etc.) (Ribas, 6 Jan 2025, Avelar et al., 2022, Martínez et al., 2021, Oh et al., 23 Nov 2025). The constant extends also to weighted and m-wise variants (Mondal et al., 2021, Zakarczemny, 2019).

2. Classical Abelian Case and Generalizations

Gao’s original setting was finite abelian groups, leveraging seminal results such as the Erdős-Ginzburg-Ziv constant: $E_1(G) = d(G) + |G| - 1$ Every sequence of $2n-1$ elements in a cyclic group $C_n$ contains a zero-sum subsequence of length $n$: $\mathsf E(C_n) = 2n-1$ For $G = C_{n_1} \oplus \cdots \oplus C_{n_r}$, with $1 < n_1 \mid n_2 \mid \cdots \mid n_r$, sharp lower and upper bounds for $d(G)$ and hence $\mathsf E(G)$ are provided (Zakarczemny, 2019): $$D^*(G) = 1+\sum_{i=1}^r (n_i-1) \leq d(G) \leq n_r\left(1+\ln\frac{|G|}{n_r}\right)$$ with exact formulas for p-groups and rank-two groups.

The m-wise generalization is established: $\mathsf E_m(G) = d(G) - 1 + m|G|$ where $\mathsf E_m(G)$ is the minimal sequence length to guarantee $m$ disjoint zero-sum blocks of size $|G|$. The proof uses zero-padding constructions and sequential extraction (Zakarczemny, 2019). Asymptotically, $\mathsf E_m(G) \sim m|G|$ as $m \to \infty$.

3. Non-Abelian Groups and Metacyclic Constructions

Gao’s constant extends to non-abelian groups, with pivotal results for metacyclic groups of the form $C_n \rtimes_s C_2$ ($s^2 \equiv 1 \mod n$). For all such groups, the exact value is now established: $\mathsf E(C_n \rtimes_s C_2) = 3n$ for $n \geq 3$ (Oh et al., 23 Nov 2025). This confirmation resolves previous obstacles, including the case $n=3n_2$ with specific divisibility and congruence properties. In these families, extremal product-one-free sequences and inverse characterizations are described explicitly.

For dihedral groups: $\mathsf E(D_{2n}) = 3n$; for $D_{2n} \times C_2$: $\mathsf E = 5n + 1$ (Martínez et al., 2021).

The proof techniques include subgroup-quotient reductions, additive-combinatorics (DeVos–Goddyn–Mohar theorem), and fine commutator analysis.

4. Weighted Variants and Jacobi Symbol Connections

Weighted zero-sum problems in cyclic groups lead to the introduction of the $A$-weighted Gao constant $E_A(n)$, defined as the minimal $k$ such that every sequence of length $k$ in $\mathbb Z_n$ admits an $A$-weighted zero-sum subsequence of length $n$ (Mondal et al., 2021). For $A$ the set of units in $\mathbb Z_n$ (i.e., $A = U(n)$), and $n$ odd and square-free, the following formula holds: $E_{U(n)}(n) = n + \Omega(n)$ where $\Omega(n)$ is the number of prime divisors of $n$. Similar extremal constructions and explicit bounds are provided for prime powers and square-free moduli.

5. Bounds, Conjectures, and Extremal Structure

The Zhuang–Gao conjecture posits universal equality $\mathsf E(G) = d(G) + |G|$ for all finite groups, though it is currently verified for wide classes (abelian, dihedral, metacyclic, nilpotent, rank-three families) (Zakarczemny, 2019, Ribas, 6 Jan 2025, Oh et al., 23 Nov 2025, Martínez et al., 2021). For non-cyclic groups, Gao–Li’s conjecture gives an upper bound (Godara et al., 2024): $\mathsf E(G) \leq 2|G|$ This bound is sharp for various semidirect products of abelian p-groups by $C_2$.

Inverse zero-sum problems often rely on characterizations of sequences of length $\mathsf E(G) - 1$ not admitting a product-one subsequence, revealing rigid structural patterns and confirming extremality in known cases.

6. Connections to Invariant Theory and Algebraic Structures

Gao’s constant is coupled with the Noether number $B(G)$ and Loewy length $L(G)$ in modular invariant theory, especially for $p$-groups and semidirect products (Godara et al., 2024). For abelian groups, $d(G)+1 = D_0(G) = B(G)$ holds, with analogous relationships in certain non-abelian families: $d(G)+1 = D_0(G) = L(G)$ where $D_0(G)$ is the ordered Davenport constant.

These equivalences establish a deep combinatorial-algebraic link, and confirm conjectures on the relationships between zero-sum invariants and the algebraic properties of the group algebra.

7. Impact and Open Problems

The determination of Gao’s constant across abelian and non-abelian groups anchors direct and inverse problems in zero-sum theory and combinatorial group theory. While the foundational cases and several large families (metacyclic, dihedral, nilpotent) are entirely classified, open questions remain for broader classes, and conjectural bounds dominate in general non-abelian contexts.

A plausible implication is continued interplay between additive combinatorics (especially extremal problems), group-theoretic invariants, and factorization theory in number fields. The structural insights offered by product-one free sequences drive both classification and construction tasks in these domains.

Gao’s constant exemplifies the confluence of combinatorial group theory with algebraic invariant theory—its study propels understanding of group-based zero-sum phenomena, the architecture of group sequences, and the efficacy of extremal combinatorial methods.