TITLE: What are the normal subgroups of a direct product?
QUESTION [11 upvotes]: Let $N$ be a normal subgroup of $G \times H$, and let $\pi_1: G \times H \to G$ and $\pi_2: G \times H \to H$ be the canonical projections. Then $\pi_1(N)$ is normal in $G$ and $\pi_2(N)$ is normal in $H$. What else can we say? I know that it is not true, in general, that $N \simeq \pi_1(N) \times \pi_2(N)$.
I'm particularly interested in the case where $G$ and $H$ are simple. In that case, $N \simeq \pi_1(N) \times \pi_2(N)$ except possibly in the case where $\pi_1(N) = G$ and $\pi_2(N) = H$. In that case, what do we know?

REPLY [5 votes]: The following corollary is from Normal Subgroup Growth of Linear Groups: the $(G_2, F_4, E_8)$-Theorem, by Michael Larsen and Alex Lubotzky.

Corollary 1.4:
  Let $G = A \times B$ be the product of two groups. If for every (finite index) normal subgroup $M \lhd A$, $Z(A/M) = \{ 1 \}$, then every (finite index) normal subgroup $N \lhd G$ is of the form $N = (N \cap A) \times (N \cap B)$.