The critical number of finite abelian groups

Abstract

Let G be an additive, finite abelian group. The critical number cr ( G ) of G is the smallest positive integer ℓ such that for every subset S ⊂ G ∖ { 0 } with | S | ⩾ ℓ 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. Erdős and H. Heilbronn in 1964, and due to the contributions of many authors the value of cr ( G ) is known for all finite abelian groups G except for G ≅ Z / p q Z where p , q are primes such that p + ⌊ 2 p − 2 ⌋ + 1 < q < 2 p . We determine that cr ( G ) = p + q − 2 for such groups.