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Let $G$ be a finite group and $H$ be a subgroup of $G$. Let $q$ be a fixed prime. Suppose that for every Sylow subgroup $P$ of $G$, the index of $H$ in $H^P$ is a power of $q$. What can we say about $|H^G:H|$?

  1. $H^X=\langle x^{-1} Hx |x\in X\rangle$

  2. I guess that $|H^G:H|$ is also a power of $q$.

  3. I can easily prove $|H^G: H|$ is a power of $q$ when $H$ is a subnormal subgroup of $G$.

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