  For a subset A of a finite abelian group G we define Sigma(A)={sum_{a\in
B}a:B\subset A}. In the case that Sigma(A) has trivial stabiliser, one may
deduce that the size of Sigma(A) is at least quadratic in |A|; the bound
|Sigma(A)|>= |A|^{2}/64 has recently been obtained by De Vos, Goddyn, Mohar and
Samal. We improve this bound to the asymptotically best possible result
|Sigma(A)|>= (1/4-o(1))|A|^{2}.
  We also study a related problem in which A is any subset of Z_{n} with all
elements of A coprime to n; it has recently been shown, by Vu, that if such a
set A has the property Sigma(A) is not Z_{n} then |A|=O(sqrt{n}). This bound
was improved to |A|<= 8sqrt{n} by De Vos, Goddyn, Mohar and Samal, we further
improve the bound to the asymptotically best possible result |A|<=
(2+o(1))sqrt{n}.
