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

We will say that an Abelian group $\mathrm{\Gamma }$ of order $n$ has the $m$-\emph{zero-sum-partition property} ($m$-$\mathit{\text{ZSP-property}}$) if $m$ divides $n$, $m\ge 2$ and there is a partition of $\mathrm{\Gamma }$ into pairwise disjoint subsets $A_1,A_2,\dots ,A_t$, such that $\mathrm{\mid }{A}_i\mathrm{\mid }=m$ and $\sum_{a\in A_i}a = g_0$for$1 \leq i \leq t$,where$g_0$istheidentityelementof$\Gamma$. In this paper we study the $m$-ZSP property of $\mathrm{\Gamma }$. We show that $\mathrm{\Gamma }$ has $m$-ZSP if and only if $\mathrm{\mid }\mathrm{\Gamma }\mathrm{\mid }$ is odd or $m\ge 3$ and $\mathrm{\Gamma }$ 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.