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**Deveno** well it's obvious that $\displaystyle C_G(H)$ is a subgroup of $\displaystyle N_G(H)$. so what we need to show is that the normalizer is a subgroup of the centralizer if |H| = 2.

so let g be in $\displaystyle N_G(H)$. clearly $\displaystyle geg^{-1} = e$, and we know that $\displaystyle ghg^{-1} \in H$, where h is the non-identity element of H.

but if $\displaystyle ghg^{-1} = e$, then gh = g, and thus h = e, contradiction. hence $\displaystyle ghg^{-1} = h \implies gh = hg$, so $\displaystyle g \in C_G(H)$