  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 \colon
V\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 the
neighborhood of $v$. Froncek %(\cite{ref_CicAus}) showed that the Cartesian
product $C_m \square C_n$, $m, n\geq3$ is a $\mathbb{Z}_{mn}$-distance magic
graph 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 labeling
for any $\alpha \equiv 0 \pmod {{\rm lcm}(m,n)}$ and any Abelian group
$\mathcal{A}$ of order $mn/\alpha$. %\cite{ref_CicAus} However, the full
characterization of group distance magic Cartesian product of two cycles is
still unknown.
  In the paper we make progress towards the complete solution this problem by
proving 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 Abelian
group $\Gamma$ of order $n^{2}$. Moreover we show that if $m\neq n$, then there
does not exist a $(\mathbb{Z}_2)^{m+n}$-distance magic labeling of the
Cartesian product $C_{2^m} \square C_{2^{n}}$. We also give necessary and
sufficient condition for $C_{m} \square C_{n}$ with $\gcd(m,n)=1$ to be
$\Gamma$-distance magic.
