Subset sums in abelian groups

Denoting by Σ(S) the set of subset sums of a subset S of a finite abelian group G, we prove that |Σ(S)|⩾|S|(|S|+2)4−1 whenever S is symmetric, |G| is odd and Σ(S) is aperiodic. Up to an additive constant of 2 this result is best possible, and we obtain the stronger (exact best possible) bound in almost all cases. We prove similar results in the case |G| is even. Our proof requires us to extend a theorem of Olson on the number of subset sums of anti-symmetric subsets S from the case of Zp to the case of a general finite abelian group. To do so, we adapt Olson's method using a generalisation of Vosper's Theorem proved by Hamidoune and Plagne.