MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $\Gamma$?

share|cite|improve this question
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
up vote 9 down vote accepted

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.

share|cite|improve this answer
Thank you very much! – Sebastian Burciu May 10 '10 at 17:23

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.