sense that and can be identified as normal in G and G''', respectively. Since is a homomorphism, its kernel N is normal in H. Moreover, given , there exists , since is surjective. Therefore, is normal in G, viz: . It follows that is normal in since . The proof that is normal in proceeds in a similar manner. Given the identification of with , we can write and instead of and , . Similarly, we can write and , . On to the proof. Consider the map defined by . The image of under this map is . Since is surjective, this relation is the graph of a well-defined function provided for every , essentially an application of the vertical line test. Since (more properly, ), we have . Thus , whence , that is, . Furthermore, for every we have . It follows that this function is a group homomorphism. By symmetry, is the graph of a well-defined homomorphism . These two homomorphisms are clearly inverse to each other and thus are indeed isomorphisms. Goursat varieties As a consequence of Goursat's theorem, one can derive a very general version on the Jordan–Hölder–Schreier theorem in Goursat varieties. References 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 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. Lemmas in group theory Articles containing proofs