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?

  • 1
    $\begingroup$ I think you should read the proof, I do not think this is suitable for exercise but everyone has different taste ... $\endgroup$
    – mesel
    Oct 6 '14 at 15:33
  • $\begingroup$ I completely agree with @Nicky, read Martin Isaacs chapter 3, It is all in there. $\endgroup$ Oct 7 '14 at 7:55
  • $\begingroup$ have you done schur zassenhaus theorem yet? $\endgroup$ Oct 7 '14 at 7:59

I advise you to read Chapter 3 (Split Extensions) of the book Finite Group Theory of I.M. Isaacs. The proofs are based on the Schur-Zassenhaus Theorem ("A finite group always splits over a normal Hall-subgroup"), of which you can appreciate the proof after having read the Hall Theorems proofs.


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