A zero-sum theorem

We determine the smallest integer n for which the following holds: if G is a nontrivial abelian group of order m, then every coloring of the integer set {1,2,…,n} by the elements of G, results a zero-sum solution to x1+x2+⋯+xm−1<xm. It turns out that n depends only on the order of G and is equal to m(m−1)+1. If G is cyclic, then we get an Erdös–Ginzburg–Ziv type generalization of a known result concerning a monochromatic solution of the above inequality in a 2-coloring of the positive integers.