On the Harborth constant of $C_3 \oplus C_{3n}$

TL;DR

This paper determines the Harborth constant for groups of the form $C_3 igoplus C_{3n}$, showing it equals $3n+3$ for prime $n eq 3$, and explicitly computes it for $C_3 igoplus C_9$ as 13.

Contribution

It provides exact values of the Harborth constant for a family of finite abelian groups, extending previous knowledge in zero-sum theory.

Findings

Harborth constant equals 3n+3 for prime n ≠ 3

Explicit value for C_3 ⊕ C_9 is 13

Generalizes zero-sum problems for specific group structures

Abstract

For a finite abelian group $(G,+, 0)$ the Harborth constant $\mathsf{g}(G)$ is the smallest integer $k$ such that each squarefree sequence over $G$ of length $k$, equivalently each subset of $G$ of cardinality at least $k$, has a subsequence of length $\exp(G)$ whose sum is $0$. In this paper, it is established that $\mathsf{g}(G)= 3n + 3$ for prime $n \neq 3$ and $\mathsf{g}(C_3 \oplus C_9)= 13$.