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 $p_{1}:H\rightarrow G$ and $p_{2}:H\rightarrow G'$ 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 $G/N\approx G'/N'$.

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

To motivate the proof, consider the slice ${\displaystyle S={g}\times G'}$ in ${\displaystyle G\times G'}$, for any arbitrary ${\displaystyle {g}\in G}$. 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 $G'$. Indeed, if we had distinct elements ${\displaystyle (g,a),(g,b)\in S}$ with ${\displaystyle a\in pN'\subset G'}$ and ${\displaystyle a\in qN'\subset G'}$ , then $H$ being a group, we get that ${\displaystyle (e,ab^{-1})\in H}$, and hence, ${\displaystyle (e,ab^{-1})\in N'}$. But this a contradiction, as $a,b$ belong to distinct cosets of $N'$, and thus ${\displaystyle ab^{-1}N'\neq N'}$, and thus the element ${\displaystyle (e,ab^{-1})\in N'}$ 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 ${\displaystyle G'\in G\times G'}$ is exactly one particular coset of ${\displaystyle N'}$ in $G'$. By an identical argument, the intersection of $H$ with every "vertical" slice isomorphic to ${\displaystyle G\in G\times G'}$ is exactly one particular coset of $N$ in $G$.

All the cosets of ${\displaystyle G,G'}$ 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 $G\times \{e'\}$ 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 $H\rightarrow G/N\times G'/N'$ defined by $(g,g')\mapsto (gN,g'N')$. The image of $H$ under this map is $\{(gN,g'N')|(g,g')\in H\}$. Since ${\displaystyle H\rightarrow G/N}$ is surjective, this relation is the graph of a well-defined function $G/N\rightarrow G'/N'$ provided ${\displaystyle g_{1}N=g_{2}N\Rightarrow g_{1}'N'=g_{2}'N'}$ for every ${\displaystyle (g_{1},g_{1}'),(g_{2},g_{2}')\in H}$, essentially an application of the vertical line test.

Since ${\displaystyle g_{1}N=g_{2}N}$ (more properly, ${\displaystyle (g_{1},e')N=(g_{2},e')N}$), we have ${\displaystyle (g_{2}^{-1}g_{1},e')\in N\subset H}$. Thus ${\displaystyle (e,g_{2}'^{-1}g_{1}')=(g_{2},g_{2}')^{-1}(g_{1},g_{1}')(g_{2}^{-1}g_{1},e')^{-1}\in H}$, whence ${\displaystyle (e,g_{2}'^{-1}g_{1}')\in N'}$, that is, ${\displaystyle g_{1}'N'=g_{2}'N'}$.

Furthermore, for every ${\displaystyle (g_{1},g_{1}'),(g_{2},g_{2}')\in H}$ we have ${\displaystyle (g_{1}g_{2},g_{1}'g_{2}')\in H}$. It follows that this function is a group homomorphism.

By symmetry, ${\displaystyle \{(g'N',gN)|(g,g')\in H\}}$ is the graph of a well-defined homomorphism ${\displaystyle G'/N'\rightarrow G/N}$. 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
• J. Lambek (1996). "The Butterfly and the Serpent". In Aldo Ursini, Paulo Agliano. Logic and Algebra. CRC Press. pp. 161–180. ISBN 978-0-8247-9606-8.
• 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.