On the products of group-magic graphs

Let A be an abelian group. We call a graph G = (V, E) A -magic if there exists a labeling f : E(G) -> A - {0} such that the induced vertex set labeling f(+) : V(G) -> A, defined by f(+) (u) = Sigma f (u, v) where the sum is over all (u, v) is an element of E(G), is a constant map. For four classical products, we examine the A -magic property of the resulting graph obtained from the product of two A-magic graphs.