Let $G$ be a connected algebraic group and $H$ a closed connected subgroup of $G$ (these are the general hypothesis I work with).
Let us denote $N_G(H) = \{ g \in G \, / \, \forall h \in H; \, ghg^{-1} \in H \}$ the Normalizer of $H$ in $G$ and $C_G(H) = \{ g \in G \, / \, \forall h \in H; \, gh = hg \}$ the Centralizer of $H$ in $G$.
Is it true that $C_G(H)$ is the identity component of $N_G(H)$ ? (i.e. the greatest connected algbraic group that contains the identity element).
Thanks in advance.
K. Y.