Unformatted text preview:  H : H ∩ P  =  HP : P  . Also  G : P  =  G : HP  HP : P  and  G : P  is not divisible by p , because P is a Sylow psubgroup of G . It follows that  H : H ∩ P  is not divisible by p , which establishes that H ∩ P is a Sylow psubgroup of H . Also it is a normal subgroup of H (for example, by the second isomorphism theorem). By Sylow’s theorems, we conclude that H ∩ P is the unique Sylow psubgroup of G ....
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 Fall '07
 PALinnell
 Math, Algebra, Group Theory, Subgroup, Homomorphism, Index of a subgroup, Sylow

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