1. The problem statement, all variables and given/known data Let G = G1 × G2 be the direct product of two simple groups. Prove that every normal subgroup of G is isomorphic to G, G1, G2, or the trivial subgroup. 3. The attempt at a solution I tried proving that the normal subgroups would have to be of the form Normal subgroup X Normal subgroup. However, that's false because, e.g., <(1,1)> is a normal subgroup of the Klein four-group.