On the Erd{\H{o}}s-Ginzburg-Ziv constant of groups of the form $C_2^r\oplus C_n$

TL;DR

This paper investigates the Erdős-Ginzburg-Ziv constant for specific finite abelian groups of the form C_2^{r-1} ⊕ C_{2n}, providing new bounds and exact values for certain cases, especially when the rank exceeds two.

Contribution

The paper introduces new upper bounds for the Erdős-Ginzburg-Ziv constant of groups C_2^{r-1} ⊕ C_{2n} with odd n and determines exact values for ranks 2 and 3.

Findings

Established a new upper bound for s(C_2^{r-1} ⊕ C_{2n}) for odd n.

Proved s(C_2^2 ⊕ C_{2n})=4n+3 for n≥2.

Proved s(C_2^3 ⊕ C_{2n})=4n+5 for n≥36.

Abstract

Let $G$ be a finite abelian group. The Erd{\H{o}}s-Ginzburg-Ziv constant $\mathsf s(G)$ of $G$ is defined as the smallest integer $l\in \mathbb{N}$ such that every sequence $S$ over $G$ of length $|S|\geq l$ has a zero-sum subsequence $T$ of length $|T|= {\exp}(G)$. The value of this classical invariant for groups with rank at most two is known. But the precise value of $\mathsf s(G)$ for the groups of rank larger than two is difficult to determine. In this paper we pay our attentions to the groups of the form $C_2^{r-1}\oplus C_{2n}$, where $r\geq 3$ and $n\ge 2$. We give a new upper bound of $\mathsf s(C_2^{r-1}\oplus C_{2n})$ for odd integer $n$. For $r\in [3,4]$, we obtain that $\mathsf s(C_2^2\oplus C_{2n})=4n+3$ for $n\ge 2$ and $\mathsf s(C_2^{3}\oplus C_{2n})=4n+5$ for $n\geq 36$.