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Let \( A \) be an abelian group. We call a graph \( G = (V, E) \) \( A \)-magic if there exists a labeling \( f : E(G) \to A^* \) such that the induced vertex set labeling \( f^+ : V(G) \to A \), defined by \( f^+(v) = \sum_{(u,v) \in E(G)} f(u,v) \), is a constant map. In this paper, we present some algebraic properties of \( A \)-magic graphs. Using them, various results are obtained for group-magic eulerian graphs.