Permutations in abelian groups and the sequence n!(modp)

We show that for any finite abelian group GG there is a permutation (g1,…,g|G|)(g1,…,g|G|) of the elements of GG such that the number of distinct sums of the form g1+⋯+gj(1≤j≤|G|) is O(|G|), and another permutation for which the number of these sums is Ω(|G|)Ω(|G|). These bounds are sharp.