Theorem $1$(Burnside): A simple nonabelian finite group can not have a conjugacy classes with prime power elemets.
Theorem $2$: A group of order $p^nq^m$ is solvable.
Theorem $1$ depends on chracter theory. Theorem $2$ is a direct consequences of theorem $1$. After Burnside Theorem $2$ is also directly proved by pure group theory.
Theorem 3: Let $G$ be a finite group with nilpotent hall $p'$ subgroup $H$. Then $G$ is solvable.
Now we can directly prove Theorem $3$ by theorem $1$. I wonder that can we prove theorem $3$ by theorem $2$ and induction as it is generelization of theorem $2$ so that theorem $3$ only depends group theory.
Thanks.