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Sylow Subgroups

  • Thread starter Obraz35
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  • #1
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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.
 

Answers and Replies

  • #2
matt grime
Science Advisor
Homework Helper
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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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