Isodynamic point

Goursat's lemma is an algebraic theorem about subgroups of the direct product of two groups.

It 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 $G/N\approx G^{\prime }/N^{\prime }$.

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

Proof of Goursat's lemma

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 $\{(gN,g^{\prime }{N}^{\prime })|(g,g^{\prime })\in H\}$. This relation is the graph of a well-defined function $G/N\to G^{\prime }/N^{\prime }$ provided $gN=N\Rightarrow g^{\prime }{N}^{\prime }=N^{\prime }$, essentially an application of the vertical line test.

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

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.