Goursat's lemma

From Wikipedia, the free encyclopedia
Jump to: navigation, search
Not to be confused with Goursat's integral lemma from Complex analysis

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 theorem also implies the snake lemma.


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

Let G, G' be groups, and let H be a subgroup of G\times G' such that the two projections p_1: H\rightarrow G and p_2: H\rightarrow G' are surjective (i.e., H is a subdirect product of G and G'). 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\times G'/N' is the graph of an isomorphism G/N\approx G'/N'.

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


Before proceeding with the proof, N and N' are shown to be normal in G \times \{e'\} and \{e\} \times 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:


It follows that N is normal in G \times \{e'\} since

(g,e')N = (g,e')(p_1(N) \times \{e'\}) = gp_1(N) \times \{e'\} = p_1(N)g \times \{e'\} = (p_1(N) \times \{e'\})(g,e')=N(g,e').

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

Given the identification of G with G \times \{e'\}, we can write G/N and gN instead of (G \times \{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. Consider the map H \rightarrow G/N \times G'/N' defined by (g,g') \mapsto (gN, g'N'). The image of H under this map is \{(gN,g'N') | (g,g') \in H \}. Since H \rightarrow G/N is surjective, this relation is the graph of a well-defined function G/N \rightarrow G'/N' provided g_1N=g_2N \Rightarrow g_1'N'=g_2'N' for every (g_1,g_1'),(g_2,g_2')\in H, essentially an application of the vertical line test.

Since g_1N=g_2N (more properly, (g_1,e')N=(g_2,e')N), we have (g_2^{-1}g_1,e') \in N \subset H. Thus (e,g_2'^{-1}g_1') = (g_2,g_2')^{-1}(g_1,g_1')(g_2^{-1}g_1,e')^{-1} \in H, whence (e,g_2'^{-1}g_1') \in N', that is, g_1'N'=g_2'N'.

Furthermore, for every (g_1,g_1'),(g_2,g_2')\in H we have (g_1g_2,g_1'g_2')\in H. It follows that this function is a group homomorphism.

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

Goursat varieties[edit]

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