# Subgroups of direct product of groups

I am interested in the following question on products of finite groups. Let $\Gamma$ be a subgroup of $U_1\times U_2$ such that the compositions with the canonical projections $\Gamma \subset U_1\times U_2 \rightarrow U_1$ and $\Gamma \subset U_1\times U_2 \rightarrow U_2$ are both surjective.

Does it follow that there is a group $G$ such that $\Gamma$ is isomorphic to the fiber product $U_1 \times_G U_2$? This means that there are surjections $\pi_1:U_1\rightarrow G$ and $\pi_2:U_2\rightarrow G$ such that $\Gamma$ is the set of pairs $(u_1,u_2)$ with $\pi_1(u_1)=\pi_2(u_2)$.

Goursat's Lemma mentioned in this question proves the statement in the case $\Gamma$ is a normal subgroup of $U_1\times U_2$.

If the statement is not true without the normality assumption, then what would be a general characterization of these subgroups $\Gamma$?

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Goursat's lemma says nothing about normality. –  Jack Schmidt May 10 '10 at 16:39
I think that your second surjection in paragraph two should be $\pi_2:U_2 \to G$. –  Sammy Black May 10 '10 at 16:42
Oh, thanks a lot! So the statement is always true. Basically $G\cong U_1/N_2 \cong U_2/N_1$ where $N_1$ and $N_2$ are the kernels of the projections $\pi_1$ and $\pi_2$. I thought that normality of $\Gamma$ is used to prove that that $N_1$ and $N_2$ are normal but from the proof from wikepedia it is clear that is not the case. –  Sebastian Burciu May 10 '10 at 16:51
@ Jack: Could you please make your comment comment as an answer? Otherwise I should perhaps answer it by myself. –  Sebastian Burciu May 10 '10 at 17:02
Goursat's Lemma also does not say anything about the canonical projections being surjective; although one can restrict to that case without loss of generality. –  José Figueroa-O'Farrill May 10 '10 at 17:08