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 $k\ge 1$. 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 $|{\sum }_n(S)|\ge k+t-1.$ This confirms a conjecture by Y.O. Hamidoune in 2000.