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 \(
W = {\left\{ {w_{i} } \right\}}^{n}_{{i = 1}}
\) is a sequence of integers whose sum is zero modulo *k*, then there exists a rearranged subsequence \(
{\left\{ {b_{i} } \right\}}^{n}_{{i = 1}}
\)of *S* such that \(
{\sum\nolimits_{i = 1}^n {w_{i} b_{i} = 0} }
\). 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. Furthermore, we in part verify a related conjecture of Y. Hamidoune, by showing that if *S* has an *n-set* partition *A*=*A*_{1}, . . .,*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*′ =*A*′_{1}, . . .,*A*′_{ n } of *S* such that \(
H \subseteq {\sum\nolimits_{i = 1}^n {w_{i} {A}\ifmmode{'}\else$'$\fi_{i} } }
\) and \(
{\left| {w_{i} {A}\ifmmode{'}\else$'$\fi_{i} } \right|} = {\left| {{A}\ifmmode{'}\else$'$\fi_{i} } \right|}
\) for all *i*, where *w*_{ i }*A*_{ i }={*w*_{ i }*a*_{ i } |*a*_{ i }∈*A*_{ i }}.

## *Mathematics Subject Classification (2000):*

11B75 05D99 ## Preview

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