If H is a subgroup of G, show that H is normal to the normalizer $\displaystyle (N(H))$

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- Oct 31st 2009, 02:04 PMsfspitfire23One more normalizer q
If H is a subgroup of G, show that H is normal to the normalizer $\displaystyle (N(H))$

- Oct 31st 2009, 02:49 PMBruno J.
$\displaystyle H$ is clearly a subgroup of $\displaystyle N_G(H)$.

To see that $\displaystyle H \lhd N_G(H)$, take any $\displaystyle x \in N_G(H)$. By definition, $\displaystyle xHx^{-1}=H$. Since this is true for all $\displaystyle x \in N_G(H)$, the statement follows.