On n-sum of an abelian group of order n

Abstract

Let $G$ be an additive finite abelian group of order $n$, and let $S$ be a sequence of $n+k$ elements in $G$, where $k\geq 1$. Suppose that $S$ contains $t$ distinct elements. Let $\sum_n(S)$ denote the set that consists of all elements in $G$ which can be expressed as the sum over a subsequence of length $n$. In this paper we prove that, either $0\in \sum_n(S)$ or $|\sum_n(S)|\geq k+t-1.$ This confirms a conjecture by Y.O. Hamidoune in 2000.