The critical number of finite abelian groups

Details The critical number of finite abelian groups The critical number of finite abelian groups

2008-10-17

arxiv.org/abs/0810.3223v1

Mathematics

Number Theory

Scientific paper

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

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