Let \(G=(V,E)\) be a finite simple graph and let \(\Gamma \) be an Abelian group of order |V|. A group distance magic labeling of G is a bijection \(\varphi :V\longrightarrow \Gamma \) with the property that there exists \(\mu \in \Gamma \) such that \(\sum _{v\in N(x)}\varphi (v)=\mu \) for any \(x\in V\) , where N(x) is the neighborhood of x and \(\mu \) is called the magic constant of this labeling. In this paper, we give a full characterization of direct product of two cycles which admits a group distance magic labeling for any finite Abelian group. This solves two conjectures proposed in Anholcer et al., [Ars. Math. Contemp. 2015].
