Summary: The main result of this paper has the following consequence. Let \(G\) be an abelian group of order \(n\). Let \(\{x_i:1\leq i\leq 2n-1\}\) be a family of elements of \(G\) and let \(\{w_i:1\leq i\leq n-1\}\) be a family of integers prime relative to \(n\). Then there is a permutation \(\tau\) of \([1,2n-1]\) such that \(\sum_{1\leq i\leq n-1}w_ix_{\tau(i)}=\sum_{1\leq i\leq n-1}w_ix_{\tau(n)}\). Applying this result with \(w_i=1\) for all \(i\), one obtains the Erdös-Ginzburg-Ziv theorem.