# Hall's Subgroup Theorem

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?

• I think you should read the proof, I do not think this is suitable for exercise but everyone has different taste ... Oct 6 '14 at 15:33
• I completely agree with @Nicky, read Martin Isaacs chapter 3, It is all in there. Oct 7 '14 at 7:55
• have you done schur zassenhaus theorem yet? Oct 7 '14 at 7:59