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Sylow Subgroups

  1. Apr 30, 2009 #1
    1. The problem statement, all variables and given/known data
    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)

    2. Relevant equations

    3. 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.
  2. jcsd
  3. Apr 30, 2009 #2

    matt grime

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    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)|
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