Exercises

1. Prove that any finite $p$-group is solvable.

   (HINT: Use Theorems 10.2.4 and 10.2.6)

2. Prove that if $G$ is a group that has a normal subgroup N such that both $N$ and $G/N$ are solvable, then $G$ must be solvable.

   (HINT: Construct the appropriate normal series for $G$ using the assumed ones for $G/N$ and $N$. Also use the Corollary to Theorem 8.3.1 and the Corollary 8.3.5.)

3. Use the original definition of solvability Definition 8.1.6) to establish Theorem 11.2.5 directly.

   (HINT: What is the image of the commutator subgroup under the canonical hom?)