The critical number of finite abelian groups

TL;DR

This paper determines the critical number for certain finite abelian groups, completing the known values for all such groups except a specific class of cyclic groups, and explicitly finds the critical number for these remaining cases.

Contribution

It explicitly computes the critical number for the remaining unresolved class of cyclic groups where the critical number was previously unknown.

Findings

Critical number is p+q-2 for groups where G ≅ Z/pqZ with p,q primes in a specific range.

Completes the determination of critical numbers for all finite abelian groups except one class.

Provides a precise value for the critical number in the unresolved cases.

Abstract

Let G be an additive, finite abelian group. The critical number $\mathsf{cr}(G)$ of $G$ is the smallest positive integer $\ell$ such that for every subset $S \subset G \setminus \{0\}$ with $|S| \ge \ell$ the following holds: Every element of $G$ can be written as a nonempty sum of distinct elements from $S$. The critical number was first studied by P. Erd\H{o}s and H. Heilbronn in 1964, and due to the contributions of many authors the value of $\mathsf {cr}(G)$ is known for all finite abelian groups $G$ except for $G \cong \mathbb{Z}/pq\mathbb{Z}$ where $p,q$ are primes such that $p+\lfloor2\sqrt{p-2}\rfloor+1<q<2p$. We determine that $\mathsf {cr}(G)=p+q-2$ for such groups.