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Let $G=(V,E)$ be a graph and $\Gamma $ an Abelian group both of order $n$. A$\Gamma$-distance magic labeling of $G$ is a bijection $\ell \colonV\rightarrow \Gamma $ for which there exists $\mu \in \Gamma $ such that $%\sum_{x\in N(v)}\ell (x)=\mu $ for all $v\in V$, where $N(v)$ is theneighborhood of $v$. Froncek %(\cite{ref_CicAus}) showed that the Cartesianproduct $C_m \square C_n$, $m, n\geq3$ is a $\mathbb{Z}_{mn}$-distance magicgraph if and only if $mn$ is even. It is also known that if $mn$ is even then$C_m \square C_n$ has $\mathbb{Z}_{\alpha}\times \mathcal{A}$-magic labelingfor any $\alpha \equiv 0 \pmod {{\rm lcm}(m,n)}$ and any Abelian group$\mathcal{A}$ of order $mn/\alpha$. %\cite{ref_CicAus} However, the fullcharacterization of group distance magic Cartesian product of two cycles isstill unknown.  In the paper we make progress towards the complete solution this problem byproving some necessary conditions. We further prove that for $n$ even the graph$C_{n}\square C_{n}$ has a $\Gamma$-distance magic labeling for any Abeliangroup $\Gamma$ of order $n^{2}$. Moreover we show that if $m\neq n$, then theredoes not exist a $(\mathbb{Z}_2)^{m+n}$-distance magic labeling of theCartesian product $C_{2^m} \square C_{2^{n}}$. We also give necessary andsufficient condition for $C_{m} \square C_{n}$ with $\gcd(m,n)=1$ to be$\Gamma$-distance magic.

Group distance magic Cartesian product of two cycles