Let and for each let be a normal subgroup. Prove that is normal and that .
We begin with some lemmas.
Lemma 1: Let and be group homomorphisms. Then . Proof: Let . Then , hence and . Thus and , and . If , then ; hence .
Lemma 2: Let and be groups with normal subgroups and . Then is normal in and . Proof: Let and denote the natural projections. These mappings are surjective, so that is a surjective homomorphism . By Lemma 1, we have . The conclusion follows by the First Isomorphism Theorem.
The main result now follows by induction on .