Proving J is Contained in a Sylow $p$-Subgroup of G

[Obraz35]

Homework Statement

If J is a subgroup of G whose order is a power of a pirme p, prove that J must be contained in a Sylow p-subgroup of G.
(Take H to be a Sylow p-subgroup of G and let X be the set of left cosets of H. Define an action of G on X by g(xH) = gxH and consider the induced action of J on X)

Homework Equations

The Attempt at a Solution

I am not sure how to begin on this one and I am also unclear on how the hint involving the group action helps in this proof.

 

[matt grime]

The only thing that ever automatically springs to mind when someone says 'consider the action of finite group K on a finite set S' is the orbit-stabilizer theorem.

|K| = |Stab(s)||Orb(s)|