Summary: An

$n$-set partition of a sequence

$S$ is a collection of

$n$ nonempty subsequences of

$S$, pairwise disjoint as sequences, such that every term of

$S$ belongs to exactly one of the subsequences, and the terms in each subsequence are all distinct so that they can be considered as sets. If S is a sequence of

$m+n-1$ elements from a finite abelian group

$G$ of order

$m$ and exponent

$k$, and if

$W={\left\{{w}_{i}\right\}}_{i=1}^{n}$ is a sequence of integers whose sum is zero modulo

$k$, then there exists a rearranged subsequence

${\left\{{b}_{i}\right\}}_{i=1}^{n}$ of S such that

${\sum}_{i=1}^{n}{w}_{i}{b}_{i}=0$. This extends the Erdős-Ginzburg-Ziv theorem, which is the case when

$m=n$ and

${w}_{i}=1$ for all

$i$, and confirms a conjecture of

*Y. Caro* [Discrete Math. 152, No. 1–3, 93–113 (1996;

Zbl 0856.05068)]. Furthermore, we in part verify a related conjecture of

*Y. O. Hamidoune* [Discrete Math. 162, No. 1–3, 127–132 (1996;

Zbl 0872.11016)] by showing that if

$S$ has an

$n$-set partition

$A={A}_{1},\cdots ,{A}_{n}$ such that

$|{w}_{i}{A}_{i}|=|{A}_{i}|$ for all

$i$, then there exists a nontrivial subgroup

$H$ of

$G$ and an

$n$-set partition

${A}^{\text{'}}={A}^{\text{'}}1,\cdots ,{A}_{n}^{\text{'}}$ of

$S$ such that

$H\subseteq {\sum}_{i=1}^{n}{w}_{i}{A}_{i}^{\text{'}}$ and

$|{w}_{i}{A}_{i}^{\text{'}}|=|{A}_{i}^{\text{'}}|$ for all

$i$, where

${w}_{i}{A}_{i}=\{{w}_{i}{a}_{i}\mid {a}_{i}\in {A}_{i}\}$.