ON A PARTITION PROBLEM OF FINITE ABELIAN GROUPS

Abstract

Let$G$be a finite abelian group and$A\subseteq G$. For$n\in G$, denote by$r_{A}(n)$the number of ordered pairs$(a_{1},a_{2})\in A^{2}$such that$a_{1}+a_{2}=n$. Among other things, we prove that for any odd number$t\geq 3$, it is not possible to partition$G$into$t$disjoint sets$A_{1},A_{2},\dots ,A_{t}$with$r_{A_{1}}=r_{A_{2}}=\cdots =r_{A_{t}}$.