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 Template:Citation needed.

Groups

Goursat's lemma for groups can be stated as follows.

Let $G$, $G^{\prime }$ be groups, and let $H$ be a subgroup of $G\times G^{\prime }$ such that the two projections $p_1:H\to G$ and $p_2:H\to G^{\prime }$ are surjective (i.e., $H$ is a subdirect product of $G$ and $G^{\prime }$). Let $N$ be the kernel of $p_2$ and $N^{\prime }$ the kernel of $p_1$. One can identify $N$ as a normal subgroup of $G$, and $N^{\prime }$ as a normal subgroup of $G^{\prime }$. Then the image of $H$ in $G/N\times G^{\prime }/N^{\prime }$ is the graph of an isomorphism $G/N\cong G^{\prime }/N^{\prime }$. One then obtains a bijection between:

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

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 $G\times G^{\prime }$ (the projections $p_1:H\to G$ and $p_2:H\to G^{\prime }$ 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 $H\le p_1(H)\times p_2(H)$.

To motivate the proof, consider the slice $S=\{g\}\times G^{\prime }$ in $G\times G^{\prime }$, 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 $N^{\prime }$. Indeed, if we have elements $(g,a),(g,b)\in S\cap H$ with $a\in p{N}^{\prime }\subset G^{\prime }$ and $b\in q{N}^{\prime }\subset G^{\prime }$, then $H$ being a group, we get that $(e,a{b}^{-1})\in H$, and hence, $(e,a{b}^{-1})\in N^{\prime }$. It follows that $(g,a)$ and $(g,b)$ lie in the same coset of $N^{\prime }$. Thus the intersection of $H$ with every "horizontal" slice isomorphic to $G^{\prime }\in G\times G^{\prime }$ is exactly one particular coset of $N^{\prime }$ in $G^{\prime }$. By an identical argument, the intersection of $H$ with every "vertical" slice isomorphic to $G\in G\times G^{\prime }$ is exactly one particular coset of $N$ in $G$.

All the cosets of $N,N^{\prime }$ 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 $N^{\prime }$ are shown to be normal in $G\times \{e^{\prime }\}$ and $\{e\}\times G^{\prime }$, respectively. It is in this sense that $N$ and $N^{\prime }$ 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 $h=(g,g^{\prime })\in H$, since $p_1$ is surjective. Therefore, $p_1(N)$ is normal in G, viz:

$g{p}_1(N)=p_1(h)p_1(N)=p_1(hN)=p_1(Nh)=p_1(N)g$.

It follows that $N$ is normal in $G\times \{e^{\prime }\}$ since

$(g,e^{\prime })N=(g,e^{\prime })(p_1(N)\times \{e^{\prime }\})=g{p}_1(N)\times \{e^{\prime }\}=p_1(N)g\times \{e^{\prime }\}=(p_1(N)\times \{e^{\prime }\})(g,e^{\prime })=N(g,e^{\prime })$.

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

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

On to the proof. Consider the map $H\to G/N\times G^{\prime }/N^{\prime }$ defined by $(g,g^{\prime })\mapsto (gN,g^{\prime }{N}^{\prime })$. The image of $H$ under this map is $\{(gN,g^{\prime }{N}^{\prime })\mid (g,g^{\prime })\in H\}$. Since $H\to G/N$ is surjective, this relation is the graph of a well-defined function $G/N\to G^{\prime }/N^{\prime }$ provided $g_{1}N=g_{2}N⟹{g_1}^{\prime }{N}^{\prime }={g_2}^{\prime }{N}^{\prime }$ for every $(g_1,{g_1}^{\prime }),(g_2,{g_2}^{\prime })\in H$, essentially an application of the vertical line test.

Since $g_{1}N=g_{2}N$ (more properly, $(g_1,e^{\prime })N=(g_2,e^{\prime })N$), we have $(g_2^{-1}{g}_1,e^{\prime })\in N\subset H$. Thus $(e,g_2{\text{'}}^{-1}{g_1}^{\prime })=(g_2,{g_2}^{\prime }{)}^{-1}(g_1,{g_1}^{\prime })(g_2^{-1}{g}_1,e^{\prime }{)}^{-1}\in H$, whence $(e,g_2{\text{'}}^{-1}{g_1}^{\prime })\in N^{\prime }$, that is, ${g_1}^{\prime }{N}^{\prime }={g_2}^{\prime }{N}^{\prime }$.

Furthermore, for every $(g_1,{g_1}^{\prime }),(g_2,{g_2}^{\prime })\in H$ we have $(g_1{g}_2,{g_1}^{\prime }{g_2}^{\prime })\in H$. It follows that this function is a group homomorphism.

By symmetry, $\{(g^{\prime }{N}^{\prime },gN)\mid (g,g^{\prime })\in H\}$ is the graph of a well-defined homomorphism $G^{\prime }/N^{\prime }\to G/N$. 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
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• 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.