In a finite group, conjugation permutes sub-Sylow subgroups

Let G be a finite group and p a prime. Prove that if H \leq G and Q \in \mathsf{Syl}_p(H), then gQg^{-1} \in \mathsf{Syl}_p(gHg^{-1}) for all g \in G.


Note that gQg^{-1} \leq gHg^{-1}. Moreover, because conjugation by g is an automorphism, |gQg^{-1}| = |Q| and |gHg^{-1}| = |H|. Hence [gHg^{-1} : gQg^{-1}] = |gHg^{-1}|/|gQg^{-1}| = |H|/|Q| = [H : Q] is not divisible by p, so that gQg^{-1} is a Sylow p-subgroup of gHg^{-1}.

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