Goursat's lemma

Goursat's lemma is an algebraic theorem.

Let $G$, $G\text{'}$ be groups, and let $H$ be a subgroup of $G\; imes\; G\text{'}$ such that the two projections $p\_1:\; H\; ightarrow\; G$ and $p\_2:\; H\; ightarrow\; G\text{'}$ are surjective. 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 $G/N\; imes\; G\text{'}/N\text{'}$ is the graph of an isomorphism $G/Napprox\; G\text{'}/N\text{'}$.

Proof of Goursat's Lemma

Before proceeding with the proof, $N$ and $N\text{'}$ are shown to be normal in $G\; imes\; \{e\text{'}\}$ and $\{e\}\; imes\; G\text{'}$, 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 $g\; in\; G$, there exists $h=(g,g\text{'})\; in\; H$, 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 $G\; imes\; \{e\text{'}\}$ since: $(g,e\text{'})N\; =\; (g,e\text{'})(p\_1(N)\; imes\; \{e\text{'}\})\; =\; gp\_1(N)\; imes\; \{e\text{'}\}\; =\; p\_1(N)g\; imes\; \{e\text{'}\}\; =\; (p\_1(N)\; imes\; \{e\text{'}\})(g,e\text{'})=N(g,e\text{'})$.

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

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

On to the proof. Let $h=(g,g\text{'})\; in\; H$. Consider the map $H\; ightarrow\; G/N\; imes\; G\text{'}/N\text{'}$ defined by $h\; mapsto\; (gN,\; g\text{'}N\text{'})$. The image of $H$ under this map is $\{(gN,g\text{'}N\text{'})\; |\; h\; in\; H\; \}$. This relation is the graph of a well-defined function $G/N\; ightarrow\; G\text{'}/N\text{'}$ provided $gN=N\; Rightarrow\; g\text{'}N\text{'}=N\text{'}$, essentially an application of the vertical line test.

Since $gN=N$ (more properly, $(g,e\text{'})N=N$), we have $(g,e\text{'})\; in\; N\; subset\; H$. Thus $(e,g\text{'})\; =\; (g,g\text{'})(g^\{-1\},e\text{'})\; in\; H$, whence $(e,g\text{'})\; in\; N\text{'}$, that is, $g\text{'}N\text{'}=N\text{'}$. Note that by symmetry, it is immediately clear that $g\text{'}N\text{'}=N\text{'}\; Rightarrow\; gN=N$, i.e., this function also passes the horizontal line test, and is therefore one-to-one. The fact that the map is a homomorphism and is surjective also follows trivially.

References

* 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.