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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    $\begingroup$ Goursat's lemma says nothing about normality. $\endgroup$ – Jack Schmidt May 10 '10 at 16:39
  • $\begingroup$ I think that your second surjection in paragraph two should be $\pi_2:U_2 \to G$. $\endgroup$ – Sammy Black May 10 '10 at 16:42
  • $\begingroup$ 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. $\endgroup$ – Sebastian Burciu May 10 '10 at 16:51
  • $\begingroup$ @ Jack: Could you please make your comment comment as an answer? Otherwise I should perhaps answer it by myself. $\endgroup$ – Sebastian Burciu May 10 '10 at 17:02
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    $\begingroup$ 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. $\endgroup$ – José Figueroa-O'Farrill May 10 '10 at 17:08

Goursat's Lemma provides a complete characterization of subgroups of a direct product of two groups as fiber products. In the language I am used to: subgroups correspond to the graphs of isomorphisms between isomorphic sections of the two factors. Some subgroup embedding properties can be read from the embedding of the sections in the factors (for instance being normal in the factors, or being central in the factors), but there are no embedding properties required to use the lemma.

Goursat's lemma appears in Roland Schmidt's Lattice of Subgroups book in chapter 1.6 (google books), and as an exercise in several textbooks.


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