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 theorem 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 $S = \ \times G'$ 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 had distinct 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 $(e, ab^) \in H$, and hence, $(e, ab^) \in N'$. But this a contradiction, as $a,b$ belong to distinct cosets of $N'$, and thus $ab^N' \neq N'$, and thus the element $(e, ab^)\in N'$ cannot belong to the kernel $N'$ of the projection map from $H$ to $G$. 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 $G,G'$ 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 \$ and $\ \times G'$, 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 \$ since
: $(g,e')N = (g,e')(p_1(N) \times \) = gp_1(N) \times \ = p_1(N)g \times \ = (p_1(N) \times \)(g,e') = N(g,e')$.

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

Given the identification of $G$ with $G \times \$, we can write $G/N$ and $gN$ instead of $(G \times \)/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 $\$. 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 $(g_2^g_1,e') \in N \subset H$. Thus $(e,g_2'^g_1') = (g_2,g_2')^(g_1,g_1')(g_2^g_1,e')^ \in H$, whence $(e,g_2'^g_1') \in N'$, 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, $\$ 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.

References

* Édouard Goursat, "Sur les substitutions orthogonales et les divisions régulières de l'espace", ''Annales Scientifiques de l'École Normale Supérieure'' (1889), Volume: 6, pages 9–102
* {{cite book, editor1=Aldo Ursini , editor2=Paulo Agliano, title=Logic and Algebra, year=1996, publisher=CRC Press, isbn=978-0-8247-9606-8, pages=161–180, author=J. Lambek, author-link=Joachim Lambek, chapter=The Butterfly and the Serpent
* Kenneth A. Ribet (Autumn 1976), " Galois Action on Division Points of Abelian Varieties with Real Multiplications", ''American Journal of Mathematics'', Vol. 98, No. 3, 751–804.

Lemmas in group theory
Articles containing proofs