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1. The problem statement, all variables and given/known data

Let G be a finite group and let primes p and q \neq p divide |G|. Prove that if G has precisely one proper Sylow p-subgroup, it is a normal subgroup, so G is simple.

EDIT: that should say "G is not simple"

2. Relevant equations

3. The attempt at a solution

I don't see the point of q. If G has precisely one proper Sylow p-subgroup, then you can conjugate with all the elements of the group and you cannot get of the subgroup or else you would have another Sylow-p-subgroup, right? So, it must be normal, right?

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# Sylow p-subgroups

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