I am currently looking into the extension of Sylow's theorems, namely through Hall-$\pi$-subgroups and Hall's Theorem. I currently have the theorem as;
Let $G$ be a finite solvable group a $\pi$ be any set of primes. Then;
$G$ has a Hall-$\pi$-subgroup,
Any two Hall-$\pi$-subgroups are conjugate,
Any subgroup whose order is a product of primes in $\pi$ is contained in some Hall-$\pi$-subgroup.
It is quite clear (to me) how these generalise the Theorems of Sylow and I understand the theorem is, in fact, an if and only if statement, but before I attempt the converse I understand Burnside's Theorem must be understood and proved.
How can I go about attempting to prove the above theorem?