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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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marked as duplicate by Anurag A, Feng Shao, Ennar, Arnaud D., Community Aug 14 at 9:13

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  • $\begingroup$ It is the very definition of the normaliser of a subgroup. $\endgroup$ – Bernard Aug 14 at 9:03