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Let $G$ be a group and $H$ a subgroup of $G$. Is there any counterexample to the assertion

$N_G(H):=\{g\in G\mid gHg^{-1}=H\}=\{g\in G\mid gHg^{-1}\subset H\}$?

Thanks!

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Let $G=GL_2(\mathbb{Q})$, let $N$ be the subgroup of upper triangular matrices with integer entries and 1's on the diagonal, and let $g$ be the diagonal matrix with entries $2,1$. Observe that $gNg^{-1} \leq N$, but $g$ does not normalize $N$. (This is an Exercise on page 88 of Dummit & Foote's Abstract Algebra text.)

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