Normal subgroups of a product of simple groups

[QIsReluctant]

Homework Statement

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.

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.

 

[micromass]

Consider the quotients $(G_1\times G_2)/ G_1$ and $(G_1\times G_2)/ G_2$.

 

[QIsReluctant]

I tried looking at the 1st isomorphism theorem with φ: (projection onto G₁), but I didn't get anywhere. Clearly the quotient groups that micromass gives are homomorphic to the simple groups ... But how would that fact lead to the conclusion that no other subgroups can exist?

I feel like I could make more progress if I understood the difference that the "simple" requirement makes ...

 

[micromass]

So you have an isomorphism $\varphi: (G_1\times G_2)/(G_1\times \{e\}) \rightarrow G_2$. Take a normal subgroup $N$ of $G_1\times G_2$. Look at $\varphi(N)$. Is this normal?