Zero sum partition into sets of the same order and its applications

Abstract

We will say that an Abelian group $\Gamma$ of order $n$ has the $m$-\emph{zero-sum-partition property} ($m$-\textit{ZSP-property}) if $m$ divides $n$, $m\geq 2$ and there is a partition of $\Gamma$ into pairwise disjoint subsets $A_1, A_2,\ldots , A_t$, such that $|A_i| = m$ and $\sum_{a\in A_i}a = g_0$ for $1 \leq i \leq t$, where $g_0$ is the identity element of $\Gamma$. In this paper we study the $m$-ZSP property of $\Gamma$. We show that $\Gamma$ has $m$-ZSP if and only if $|\Gamma|$ is odd or $m\geq 3$ and $\Gamma$ has more than one involution. We will apply the results to the study of group distance magic graphs as well as to generalized Kotzig arrays.