Goursat's lemma

Goursat's lemma is an algebraic theorem.

Let $$G, $$G' be groups, and let $$H be a subgroup of $$G imes G' such that the two projections $$p_1: H ightarrow G and $$p_2: H ightarrow G' are surjective. Let $$N be the kernel of $$p_2 and $$N' the kernel of $$p_1. One can identify $$N as a normal subgroup of $$G, and $$N' as a normal subgroup of $$G'. Then the image of $$H in $$G/N imes G'/N' is the graph of an isomorphism $$G/Napprox G'/N'.

Proof of Goursat's Lemma

Before proceeding with the proof, $$N and $$N' are shown to be normal in $$G imes {e'} and $${e} imes G', respectively. It is in this sense that $$N and $$N' 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 $$h=(g,g') in H, since $$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'} since: $$(g,e')N = (g,e')(p_1(N) imes {e'}) = gp_1(N) imes {e'} = p_1(N)g imes {e'} = (p_1(N) imes {e'})(g,e')=N(g,e').

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

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

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

Since $$gN=N (more properly, $$(g,e')N=N), we have $$(g,e') in N subset H. Thus $$(e,g') = (g,g')(g^{-1},e') in H, whence $$(e,g') in N', that is, $$g'N'=N'. Note that by symmetry, it is immediately clear that $$g'N'=N' 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.