Goursat's lemma

Not to be confused with Goursat's integral lemma from complex analysis.

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 ${\displaystyle G}$, ${\displaystyle G'}$ be groups, and let ${\displaystyle H}$ be a subgroup of ${\displaystyle G\times G'}$ such that the two projections ${\displaystyle p_{1}:H\to G}$ and ${\displaystyle p_{2}:H\to G'}$ are surjective (i.e., ${\displaystyle H}$ is a subdirect product of ${\displaystyle G}$ and ${\displaystyle G'}$). Let ${\displaystyle N}$ be the kernel of ${\displaystyle p_{2}}$ and ${\displaystyle N'}$ the kernel of ${\displaystyle p_{1}}$. One can identify ${\displaystyle N}$ as a normal subgroup of ${\displaystyle G}$, and ${\displaystyle N'}$ as a normal subgroup of ${\displaystyle G'}$. Then the image of ${\displaystyle H}$ in ${\displaystyle G/N\times G'/N'}$ is the graph of an isomorphism ${\displaystyle G/N\cong G'/N'}$. One then obtains a bijection between:

1. Subgroups of ${\displaystyle G\times G'}$ which project onto both factors,
2. Triples ${\displaystyle (N,N',f)}$ with ${\displaystyle N}$ normal in ${\displaystyle G}$, ${\displaystyle N'}$ normal in ${\displaystyle G'}$ and ${\displaystyle f}$ isomorphism of ${\displaystyle G/N}$ onto ${\displaystyle G'/N'}$.

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 ${\displaystyle H}$ is any subgroup of ${\displaystyle G\times G'}$ (the projections ${\displaystyle p_{1}:H\to G}$ and ${\displaystyle p_{2}:H\to G'}$ need not be surjective), then the projections from ${\displaystyle H}$ onto ${\displaystyle p_{1}(H)}$ and ${\displaystyle p_{2}(H)}$ are surjective. Then one can apply Goursat's lemma to ${\displaystyle H\leq p_{1}(H)\times p_{2}(H)}$.

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

All the cosets of ${\displaystyle N,N'}$ are present in the group ${\displaystyle H}$, 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, ${\displaystyle N}$ and ${\displaystyle N'}$ are shown to be normal in ${\displaystyle G\times \{e'\}}$ and ${\displaystyle \{e\}\times G'}$, respectively. It is in this sense that ${\displaystyle N}$ and ${\displaystyle N'}$ can be identified as normal in G and G', respectively.

Since ${\displaystyle p_{2}}$ is a homomorphism, its kernel N is normal in H. Moreover, given ${\displaystyle g\in G}$, there exists ${\displaystyle h=(g,g')\in H}$, since ${\displaystyle p_{1}}$ is surjective. Therefore, ${\displaystyle p_{1}(N)}$ is normal in G, viz:

${\displaystyle gp_{1}(N)=p_{1}(h)p_{1}(N)=p_{1}(hN)=p_{1}(Nh)=p_{1}(N)g}$.

It follows that ${\displaystyle N}$ is normal in ${\displaystyle G\times \{e'\}}$ since

${\displaystyle (g,e')N=(g,e')(p_{1}(N)\times \{e'\})=gp_{1}(N)\times \{e'\}=p_{1}(N)g\times \{e'\}=(p_{1}(N)\times \{e'\})(g,e')=N(g,e')}$.

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

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

On to the proof. Consider the map ${\displaystyle H\to G/N\times G'/N'}$ defined by ${\displaystyle (g,g')\mapsto (gN,g'N')}$. The image of ${\displaystyle H}$ under this map is ${\displaystyle \{(gN,g'N')\mid (g,g')\in H\}}$. Since ${\displaystyle H\to G/N}$ is surjective, this relation is the graph of a well-defined function ${\displaystyle G/N\to G'/N'}$ provided ${\displaystyle g_{1}N=g_{2}N\implies g_{1}'N'=g_{2}'N'}$ 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, ${\displaystyle \{(g'N',gN)\mid (g,g')\in H\}}$ is the graph of a well-defined homomorphism ${\displaystyle G'/N'\to G/N}$. 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
• J. Lambek (1996). "The Butterfly and the Serpent". In Aldo Ursini; Paulo Agliano (eds.). 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.
• A. Carboni, G.M. Kelly and M.C. Pedicchio (1993), Some remarks on Mal'tsev and Goursat categories, Applied Categorical Structures, Vol. 4, 385–421.