Let $H$ be a subgroup of $G$.
$S=\{g\in G\mid ghg^{-1}\in H,\forall h\in H\}$.
I know that $N_G(H)\subset S$. But I can't seem to find any counterexamples for the other inclusion. Any ideas?
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It is the very definition of the normaliser of a subgroup. – Bernard Aug 14 '19 at 09:03
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Note that if $gHg^{-1} \subset H$, then actually $gHg^{-1} = H$. – D_S Dec 09 '19 at 05:05