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 $\Gamma$?
Best Answer
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.