Covering a finite abelian group by subset sums (Q1882116)

Let \(G\) be an finite additive abelian group of order \(n\geq 3\), and let \(S\) be a subset of non-zero elements of \(G\). Then \(\sum(S)=\{\sum_{x\in A} x\mid A\subseteq S,\;A\neq \emptyset\}\) is the set of all subset sums of \(S\). The critical number \(c(G)\) of \(G\) is the smallest \(s\) such that the subset sums of \(\sum(S)\) cover all \(G\) for each subset \(S\subset G\{0\} \) of cardinality \(| S| \geq s\). Some years ago it has been proved that, if \(p\) is the smallest prime dividing \(n\) and \(n/p=h\) is composite, then \(c(G)=h+p-2\). In this paper the sets with \(| S| =h+p-3\) and \(\sum(S)=G\) are characterized, where \(p\geq 3\) is the smallest prime dividing \(n\), \(n/p=h\) is composite and \(n\geq 7p^2+3p\). There are also some further results for groups with odd order.