Let G be an abelian group of order n and let μ be a sequence of elements of G with length 2n−k+1 taking k distinct values. Assuming that no value occurs n−k+3 times, we prove that the sums of the n-subsequences of μ must include a non-null subgroup. As a corollary we show that if G is cyclic then μ has an n-subsequence summing to 0. This last result, conjectured by Bialostocki, reduces to the Erdos–Ginzburg–Ziv theorem for k=2.

Let G be a ÿnite abelian group. A sequence in G is barycentric if it contains one element which is the "average" of its terms. We give some su cient conditions for the existence of barycentric sequences, with prescribed or unconstrained length.