### Zero Sum Partition of Abelian Groups into Sets of the Same Order and its Applications

#### Abstract

We will say that an Abelian group $\Gamma$ of order $n$ has the $m$-*zero-sum-partition property* ($m$-*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 the $m$-ZSP property if and only if $m\geq 3$ and $|\Gamma|$ is odd or $\Gamma$ 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.

#### Keywords

Abelian group; Zero sum partition; Group distance magic labelling; Kotzig arrays