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 (S) covers all G for eachs ubset SG\{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 (S)=G, where p3 is the smallest prime dividing n, n/p is composite and n7p²+3p.We also extend a result of Diderrichan d Mann by proving that, for n67, |S|n/3+2 and S=G imply (S)=G. Sets of cardinality for which (S) =G are also characterized when n183, the smallest prime p dividing n is odd and n/p is composite. Finally we obtain a necessary and sufficient condition for the equality (G)=G to hold when |S|n/(p+2)+p, where p5, n/p is composite and n15p².* Work partially supported by the Spanish Research Council under grant TIC2000-1017 Work partially supported by the Catalan Research Council under grant 2000SGR00079