Covering a finite Abelian group by subset sums

Let G be an abelian group of order n. The critical number c(G) of C is the smallest s such that the subset sums set Sigma(S) covers all G for each subset S subset of G\{0} of cardinality \S\ greater than or equal to 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 Sigma(S) = G, where p greater than or equal to 3 is the smallest prime dividing n, n/p is composite and n greater than or equal to 7 p(2) + 3p. We also extend a result of Diderrich and Mann by proving that, for n greater than or equal to 67, \S\ > n/3+2 and [S] = G imply Sigma(S) = G. Sets of cardinality \S\ greater than or equal to n+11/4 for which Sigma(S) not equal G are also characterized when n greater than or equal to 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 Sigma(G) = G to hold when \S\ greater than or equal to n/(p+ 2) +p, where p greater than or equal to 5, n/p is composite and n greater than or equal to 15 p(2).