Covering a Finite Abelian Group by Subset Sums

Let G be an abelian group of order n. The critical number c(G) of G is the smallest s such that the subset sums set Σ(S) covers all G for eachs ubset S⊂G\{0} of cardinality |S|≥s. It has been recently proved that, if p is the smallest prime dividing n and n/p is composite, then c(G)=|G|/p+p−2, thus establishing a conjecture of Diderrich.

We characterize the critical sets with |S|=|G|/p+p−3 and Σ(S)=G, where p≥3 is the smallest prime dividing n, n/p is composite and n≥7p ²+3p.

We also extend a result of Diderrichan d Mann by proving that, for n≥67, |S|≥n/3+2 and S=G imply Σ(S)=G. Sets of cardinality ${\left| S \right|} \geqslant \frac{{n + 11}} {4}$ for which Σ(S) =G are also characterized when n≥183, the smallest prime p dividing n is odd and n/p is composite. Finally we obtain a necessary and sufficient condition for the equality Σ(G)=G to hold when |S|≥n/(p+2)+p, where p≥5, n/p is composite and n≥15p ².