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
}}.

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Grynkiewicz, D.J. A Weighted Erdős-Ginzburg-Ziv Theorem.
*Combinatorica* **26, **445–453 (2006). https://doi.org/10.1007/s00493-006-0025-y

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*Mathematics Subject Classification (2000):*

- 11B75
- 05D99