Permutations in abelian groups and the sequence

We show that for any finite abelian group G there is a permutation ( g 1 , … , g | G | ) of the elements of G such that the number of distinct sums of the form g 1 + ⋯ + g j ( 1 ≤ j ≤ | G | ) is O ( | G | ) , and another permutation for which the number of these sums is Ω ( | G | ) . These bounds are sharp.