For any abelian group \*A\), we call a graph \(G = (V, E)\) as A-magic if there exists a labeling I: E(G) \(\to \text{A} – \{0\}\) such that the induced vertex set labeling \(I^+: V(G) \to A\)
\[\text{I}^+\text{(v)} = \Sigma \{ \text{I(u,v) : (u,v) in E(G)} \}\]
is a constant map. We denote the set of all \(A\) such that G is \(A\)-magic by \(AM(G)\) and call it as group-magic index set of \(G\).