We are going to reconstruct the salient of a part of the classical theorem:
Each p-S of is contained into a p-SS of
Here the reference frame:
1. p-SS divides :
The set p-SS must be considered as an orbit of the action p-SS p-SS via . Since from sylow II we know that each two are conjugated the there is only one orbit
giving us a trivial orbital partition. Then p-SS=, i.e.
because for the isotropy group is .
2. Observe also that #p-SS:
since and , as far as
which implies that p-SS, then p-SS.
Proof of Theorem:
Considering the conjugation sub-action
we get a orbit decomposition p-SS = with the corresponding class equation
#p-SS = ,
but asumming that then
#p-SS = .
Now since #p-SS then there is some for which , so
With that, it is easy these: and is normal in .
Also given by is an epimorphism, so by the fundamental theorem of group-morphisms we have
Observing that and then
but is maximal, then . Hence and