Goursat's lemma

Goursat's lemma, named after the French mathematician Édouard Goursat, is an algebraic theorem about subgroups of the direct product of two groups.

It can be stated more generally in a Goursat variety (and consequently it also holds in any Maltsev variety), from which one recovers a more general version of Zassenhaus' butterfly lemma. In this form, Goursat's lemma also implies the snake lemma.

Groups
Goursat's lemma for groups can be stated as follows.
 * 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 (i.e., $$H$$ is a subdirect product of $$G$$ and $$G'$$). Let $$N$$ be the kernel of $$p_2$$ and $$N'$$ 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 $$G/N \cong G'/N'$$. One then obtains a bijection between:
 * Subgroups of $$G\times G'$$ which project onto both factors,
 * Triples $$(N, N', f)$$ with $$N$$ normal in $$G$$, $$N'$$ normal in $$G'$$ and $$f$$ isomorphism of $$G/N$$ onto $$G'/N'$$.

An immediate consequence of this is that the subdirect product of two groups can be described as a fiber product and vice versa.

Notice that if $$H$$ is any subgroup of $$G\times G'$$ (the projections $$p_1: H \to G$$ and $$p_2: H \to G'$$ need not be surjective), then the projections from $$H$$ onto $$p_1(H)$$ and $$ p_2(H)$$ are surjective. Then one can apply Goursat's lemma to $$H \leq p_1(H)\times p_2(H)$$.

To motivate the proof, consider the slice $$S = \{g\} \times G'$$ in $$G \times G'$$, for any arbitrary $$g \in G$$. By the surjectivity of the projection map to $$G$$, this has a non trivial intersection with $$H$$. Then essentially, this intersection represents exactly one particular coset of $$N'$$. Indeed, if we have elements $$(g,a), (g,b) \in S \cap H$$ with $$a \in pN' \subset G'$$ and $$b \in qN' \subset G'$$, then $$H$$ being a group, we get that $$(e, ab^{-1}) \in H$$, and hence, $$(e, ab^{-1}) \in N'$$. It follows that $$(g,a)$$ and $$(g,b)$$ lie in the same coset of $$N'$$. Thus the intersection of $$H$$ with every "horizontal" slice isomorphic to $$G' \in G\times G'$$ is exactly one particular coset of $$N'$$ in $$G'$$. By an identical argument, the intersection of $$H$$ with every "vertical" slice isomorphic to $$G \in G\times G'$$ is exactly one particular coset of $$N $$ in $$G$$.

All the cosets of $$N,N'$$ are present in the group $$H$$, and by the above argument, there is an exact 1:1 correspondence between them. The proof below further shows that the map is an isomorphism.

Proof
Before proceeding with the proof, $$N$$ and $$N'$$ are shown to be normal in $$G \times \{e'\}$$ and $$\{e\} \times G'$$, respectively. It is in this sense that $$N$$ and $$N'$$ can be identified as normal in G and G', respectively.

Since $$p_2$$ is a homomorphism, its kernel N is normal in H. Moreover, given $$g \in G$$, there exists $$h=(g,g') \in H$$, since $$p_1$$ is surjective. Therefore, $$p_1(N)$$ is normal in G, viz:
 * $$gp_1(N) = p_1(h)p_1(N) = p_1(hN) = p_1(Nh) = p_1(N)g$$.

It follows that $$N$$ is normal in $$G \times \{e'\}$$ since
 * $$(g,e')N = (g,e')(p_1(N) \times \{e'\}) = gp_1(N) \times \{e'\} = p_1(N)g \times \{e'\} = (p_1(N) \times \{e'\})(g,e') = N(g,e')$$.

The proof that $$N'$$ is normal in $$\{e\} \times G'$$ proceeds in a similar manner.

Given the identification of $$G$$ with $$G \times \{e'\}$$, we can write $$G/N$$ and $$gN$$ instead of $$(G \times \{e'\})/N$$ and $$(g,e')N$$, $$g \in G$$. Similarly, we can write $$G'/N'$$ and $$g'N'$$, $$g' \in G'$$.

On to the proof. Consider the map $$H \to G/N \times G'/N'$$ defined by $$(g,g') \mapsto (gN, g'N')$$. The image of $$H$$ under this map is $$\{(gN,g'N') \mid (g,g') \in H \}$$. Since $$H \to G/N$$ is surjective, this relation is the graph of a well-defined function $$G/N \to G'/N'$$ provided $$g_1N = g_2N \implies g_1'N' = g_2'N'$$ for every $$(g_1,g_1'),(g_2,g_2') \in H$$, essentially an application of the vertical line test.

Since $$g_1N=g_2N$$ (more properly, $$(g_1,e')N = (g_2,e')N$$), we have $$(g_2^{-1}g_1,e') \in N \subset H$$. Thus $$(e,g_2'^{-1}g_1') = (g_2,g_2')^{-1}(g_1,g_1')(g_2^{-1}g_1,e')^{-1} \in H$$, whence $$(e,g_2'^{-1}g_1') \in N'$$, that is, $$g_1'N'=g_2'N'$$.

Furthermore, for every $$(g_1,g_1'),(g_2,g_2')\in H$$ we have $$(g_1g_2,g_1'g_2')\in H$$. It follows that this function is a group homomorphism.

By symmetry, $$\{(g'N',gN) \mid (g,g') \in H \}$$ is the graph of a well-defined homomorphism $$G'/N' \to G/N$$. These two homomorphisms are clearly inverse to each other and thus are indeed isomorphisms.

Goursat varieties
As a consequence of Goursat's theorem, one can derive a very general version on the Jordan–Hölder–Schreier theorem in Goursat varieties.