If $H$ is a subgroup of a group $ G$. Then the normalizer of $H$ is defined as,
$N_G(H)=${$g\in G: gHg^{-1}=H$}.
However, $gHg^{-1}=H$ if and only if $gH=Hg$, which implies that $H$ is a normal subgroup of $G$ (since left and right cosets of $H$ are identical). Is this observation correct? Please someone help me out. Thanks in advance.
Best Answer
You've just noticed that a subgroup is a normal subgroup of its normalizer.
This is, in a sense, the whole idea of the normalizer. In fact, as @Nicky Hekster observes, the normalizer is the largest such subgroup.