Prove that subgroups and quotient groups of solvable groups are solvable.
We prove some lemmas first.
Lemma 1: Let be a group and let with normal in . Then is normal in . Proof: Let . Then because and .
Lemma 2: Let , , and be groups with normal in . Then is normal in and there is an injective homomorphism . Proof: Let be the natural projection. Certainly , and if , we have , and so . By the First Isomorphism Theorem, the induced homomorphism is injective.
Lemma 3: Let be a group and such that is normal in and is normal, and . Then is normal. Proof: Let . Then .
Let be a solvable group. Then there exists a subnormal series such that is abelian for all .
Let be a subgroup. By Lemma 1 we have . Moreover, by Lemma 2 we have abelian. Thus is solvable.
Let be a normal subgroup. Now is normal by Lemma 3, so that . Moreover, is abelian by the Third Isomorphism Theorem. Hence is solvable.