TITLE: Goursat's Lemma proof
QUESTION [5 upvotes]: There is a problem in Lang's book that I don't quite understand how to proceed. It is problem #5, pg 75. I  have already shown that the subgroups N and N' can be identified as normal in G, G'. But I don't know how to show that the image of H in G/N$\times$G'/N' is the graph of an isomorphism.
Problem statement:

Let $G, G'$ be groups, and let $H$ be a subgroup of $G \times G'$ such that the two projections $p_1 : H \to G$ and $p_2 : H \to G'$ are surjective. Let $N$ be the kernel of $p_2$ and $N'$ be the kernel of $p_1$.
One can identify $N$ as a normal subgroup of $G$, and $N'$ as a normal subgroup of $G'$. Then the image of $H$ in $G/N \times G'/N'$ is the graph of an isomorphism
$
G/N \approx G'/N'
$.

REPLY [5 votes]: It seems to me that you're at a stage where you don't know what you need to try proving, so instead of giving a full answer, I'll give a hint that will hopefully clarify things.
Prove the following lemma: Given groups $H, K, K'$ and surjective morphisms $H \to K$ and $H \to K'$ with the same kernel, the image of the induced morphism $H \to K \times K'$ is the graph of an isomorphism between $K$ and $K'$. (Note that having the same kernel is actually a necessary condition, since no elements $(1,k')$ or $(k,1)$ are permitted to appear in the image.) To prove the lemma, it may be helpful to factor first through the common kernel $M$, at which point it should become obvious by identifying each factor with $H/M$.
Now return to the original situation and see what needs to be proved.