The Erdős-Ginzburg-Ziv theorem constant of finite groups

Let $G$ be a multiplicatively written finite group of order $n$. The Erdős-Ginzburg-Ziv Theorem constant of the group $G$, denoted $\mathsf E(G)$, is defined as the smallest positive integer $\ell$ with the following property: for any given sequence $(g_1,\ldots,g_{\ell})$ over $G$, there exist $n$ distinct integers $i_1,\ldots,i_n\in {1,\ldots,\ell}$ such that the product of $g_{i_1},\ldots,g_{i_n}$, in some order, is the identity element of $G$. The Erdős-Ginzburg-Ziv Theorem constant originates from the celebrated additive theorem proved by Erdős, Ginzburg and Ziv in 1961, which amounts to proving $\mathsf E(G)\leq 2|G|-1$ holds in case that $G$ is abelian. It is also well-known that $\mathsf E(G)=2|G|-1$ holds for all finite cyclic groups. In 2010, Gao and Li [J. Pure Appl. Algebra] conjectured that $\mathsf E(G)\leq \frac{3|G|}{2}$ for every finite non-cyclic group $G$. In this paper, we confirm the conjecture for all non-cyclic groups $G$ whose order is not divisible by four, and characterize the groups achieving the equality $\mathsf E(G)=\frac{3|G|}{2}$ as those with a cyclic subgroup of index two.

💡 Research Summary

The paper investigates the Erdős‑Ginzburg‑Ziv (EGZ) constant E(G) for finite groups G, defined as the smallest integer ℓ such that every sequence of ℓ elements from G contains a subsequence of length |G| whose product (in some order) equals the identity. While the classical EGZ theorem guarantees E(G) ≤ 2|G| − 1 for abelian groups, and Olson extended this bound to all finite groups, the authors focus on a conjecture of Gao and Li (2010) stating that for any non‑cyclic finite group, E(G) ≤ 3|G|/2.

The main result (Theorem 1.2) confirms this conjecture for all non‑cyclic groups whose order is not divisible by four. Moreover, the paper characterizes precisely when equality holds: E(G) = 3|G|/2 if and only if G possesses a cyclic subgroup of index two (i.e., a normal cyclic subgroup of order |G|/2).

The proof proceeds through several layers. First, Lemma 3.1 (a result of Halter‑Koch) gives an upper bound for the small Davenport constant d(G): d(G) ≤ |G|/p + p − 2, where p is the smallest prime divisor of |G|. Lemma 3.2 shows that E(G) ≥ d(G)+|G|, while Lemma 3.3 asserts equality for abelian groups. Lemma 3.5 provides a lifting argument: if a normal subgroup H satisfies E(G) ≤ c|H| − 1 for some 1 < c ≤ 2, then the same inequality holds for G. Lemma 3.6 (a classic group‑theoretic fact) guarantees that any group of order 2m with m odd contains a subgroup of index two.

Using these tools, the authors treat the case where |G| is even but not divisible by four. They first handle metacyclic groups C_m ⋊ C_2 (Lemma 3.8), showing that for such groups E(G)=3|G|/2. They then analyze groups isomorphic to S_3×C_3 or (C_3×C_3)⋊C_2 (Lemma 3.11), establishing the bound E(G) ≤ 3|G|/2 − 1.

A substantial part of the argument is a delicate combinatorial decomposition of a hypothetical extremal sequence S of length 3|G|/2 that lacks a product‑one subsequence of length |G|. The sequence is partitioned into short blocks T_i of length two whose products lie in a normal subgroup H of index two, together with a possible remainder W. By projecting onto the quotient G/H (which is C_2) and using the known value E(C_2)=3, the authors control the number of such blocks. They then construct a derived sequence \tilde S in H consisting of the block products and apply Lemma 3.9, a structural description of long product‑one‑free sequences over C_n×C_n, to bound the multiplicities of elements in \tilde S. This yields two possible configurations for \tilde S, each leading to contradictions after a careful case analysis (r≥1 and r=0, where r counts blocks lying outside H). The contradictions rely on the fact that H≅C_3×C_3 has Davenport constant d(H)=4, forcing any sufficiently long subsequence of the remainder to contain a product‑one subsequence, which ultimately produces a forbidden product‑one subsequence of length |G| in the original S.

Consequently, no counterexample exists, and the bound E(G) ≤ 3|G|/2 holds for all non‑cyclic groups with 4∤|G|. The equality case follows from Lemma 3.8 and the subgroup‑of‑index‑two characterization: if G has a cyclic subgroup of index two, then the construction in Lemma 3.8 shows that the bound is tight; conversely, if equality holds, the analysis forces the existence of such a subgroup.

The paper thus resolves the Gao‑Li conjecture in the substantial class of groups whose order is not a multiple of four, improves the general EGZ bound from 2|G| − 1 to 3|G|/2 for these groups, and provides a complete structural description of the extremal cases. The techniques blend additive combinatorics (Davenport constants, product‑one free sequences) with classical group theory (normal subgroups of index two, metacyclic presentations), offering a template for further investigations into EGZ constants of more general non‑abelian groups.