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 Z p 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.