A group homomorphism is a map f:G->H between two groups such that the group operation is preserved:f(g_1 g_2) = f(g_1) f(g_2) for all g_1, g_2 element G, where the product on the left-hand side is in G and on the right-hand side in H. As a result, a group homomorphism maps the identity element in G to the identity element in H: f(e_G) = e_H. Note that a homomorphism must preserve the inverse map because f(g) f(g^(-1)) = f(g g^(-1)) = f(e_G) = e_H, so f(g)^(-1) = f(g^(-1)).
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