Goursat's lemma

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

: Let $G$, $G\text{'}$ be groups, and let $H$ be a subgroup of $G\backslash times\; G\text{'}$ such that the two projections $p\_1:\; H\; \backslash to\; G$ and $p\_2:\; H\; \backslash to\; G\text{'}$ are surjective (i.e., $H$ is a subdirect product of $G$ and $G\text{'}$). Let $N$ be the kernel of $p\_2$ and $N\text{'}$ the kernel of $p\_1$. One can identify $N$ as a normal subgroup of $G$, and $N\text{'}$ as a normal subgroup of $G\text{'}$. Then the image of $H$ in $G/N\; \backslash times\; G\text{'}/N\text{'}$ is the graph of an isomorphism $G/N\; \backslash cong\; G\text{'}/N\text{'}$. One then obtains a bijection between:

:# Subgroups of $G\backslash times\; G\text{'}$ which project onto both factors,

:# Triples $(N,\; N\text{'},\; f)$ with $N$ normal in $G$, $N\text{'}$ normal in $G\text{'}$ and $f$ isomorphism of $G/N$ onto $G\text{'}/N\text{'}$.

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 $G\backslash times\; G\text{'}$ (the projections $p\_1:\; H\; \backslash to\; G$ and $p\_2:\; H\; \backslash to\; G\text{'}$ 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 $H\; \backslash leq\; p\_1(H)\backslash times\; p\_2(H)$.

To motivate the proof, consider the slice $S\; =\; \backslash \{g\backslash \}\; \backslash times\; G\text{'}$ in $G\; \backslash times\; G\text{'}$, for any arbitrary $g\; \backslash in\; G$. 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 $N\text{'}$. Indeed, if we have elements $(g,a),\; (g,b)\; \backslash in\; S\; \backslash cap\; H$ with $a\; \backslash in\; pN\text{'}\; \backslash subset\; G\text{'}$ and $b\; \backslash in\; qN\text{'}\; \backslash subset\; G\text{'}$, then $H$ being a group, we get that $(e,\; ab^\{-1\})\; \backslash in\; H$, and hence, $(e,\; ab^\{-1\})\; \backslash in\; N\text{'}$. It follows that $(g,a)$ and $(g,b)$ lie in the same coset of $N\text{'}$. Thus the intersection of $H$ with every "horizontal" slice isomorphic to $G\text{'}\; \backslash in\; G\backslash times\; G\text{'}$ is exactly one particular coset of $N\text{'}$ in $G\text{'}$.

By an identical argument, the intersection of $H$ with every "vertical" slice isomorphic to $G\; \backslash in\; G\backslash times\; G\text{'}$ is exactly one particular coset of $N$ in $G$.

All the cosets of $N,N\text{'}$ 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.

Before proceeding with the proof, $N$ and $N\text{'}$ are shown to be normal in $G\; \backslash times\; \backslash \{e\text{'}\backslash \}$ and $\backslash \{e\backslash \}\; \backslash times\; G\text{'}$, respectively. It is in this sense that $N$ and $N\text{'}$ 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 $g\; \backslash in\; G$, there exists $h=(g,g\text{'})\; \backslash in\; H$, 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 $G\; \backslash times\; \backslash \{e\text{'}\backslash \}$ since

: $(g,e\text{'})N\; =\; (g,e\text{'})(p\_1(N)\; \backslash times\; \backslash \{e\text{'}\backslash \})\; =\; gp\_1(N)\; \backslash times\; \backslash \{e\text{'}\backslash \}\; =\; p\_1(N)g\; \backslash times\; \backslash \{e\text{'}\backslash \}\; =\; (p\_1(N)\; \backslash times\; \backslash \{e\text{'}\backslash \})(g,e\text{'})\; =\; N(g,e\text{'})$.

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

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

On to the proof. Consider the map $H\; \backslash to\; G/N\; \backslash times\; G\text{'}/N\text{'}$ defined by $(g,g\text{'})\; \backslash mapsto\; (gN,\; g\text{'}N\text{'})$. The image of $H$ under this map is $\backslash \{(gN,g\text{'}N\text{'})\; \backslash mid\; (g,g\text{'})\; \backslash in\; H\; \backslash \}$. Since $H\; \backslash to\; G/N$ is surjective, this relation is the graph of a well-defined function $G/N\; \backslash to\; G\text{'}/N\text{'}$ provided $g\_1N\; =\; g\_2N\; \backslash implies\; g\_1\text{'}N\text{'}\; =\; g\_2\text{'}N\text{'}$ for every $(g\_1,g\_1\text{'}),(g\_2,g\_2\text{'})\; \backslash in\; H$, essentially an application of the vertical line test.

Since $g\_1N=g\_2N$ (more properly, $(g\_1,e\text{'})N\; =\; (g\_2,e\text{'})N$), we have $(g\_2^\{-1\}g\_1,e\text{'})\; \backslash in\; N\; \backslash subset\; H$. Thus $(e,g\_2\text{'}^\{-1\}g\_1\text{'})\; =\; (g\_2,g\_2\text{'})^\{-1\}(g\_1,g\_1\text{'})(g\_2^\{-1\}g\_1,e\text{'})^\{-1\}\; \backslash in\; H$, whence $(e,g\_2\text{'}^\{-1\}g\_1\text{'})\; \backslash in\; N\text{'}$, that is, $g\_1\text{'}N\text{'}=g\_2\text{'}N\text{'}$.

Furthermore, for every $(g\_1,g\_1\text{'}),(g\_2,g\_2\text{'})\backslash in\; H$ we have $(g\_1g\_2,g\_1\text{'}g\_2\text{'})\backslash in\; H$. It follows that this function is a group homomorphism.

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

{{expand section|date=April 2015}}

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

• 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.
• 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.

Category:Lemmas in group theory

Category:Articles containing proofs