# Every group is normal fully normalized in its holomorph

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Suppose $G$ is a group. Denote by $\operatorname{Hol}(G)$ the holomorph of $G$; in other words, we have:
$\operatorname{Hol}(G) = G \rtimes \operatorname{Aut}(G)$
with the natural action of the automorphism group $\operatorname{Aut}(G)$ on $G$. Then, $G$ is a normal fully normalized subgroup of $\operatorname{Hol}(G)$:
• $G$ is normal in $\operatorname{Hol}(G)$: Every inner automorphism of $\operatorname{Hol}(G)$ restricts to an automorphism of $G$.
• $G$ is fully normalized in $\operatorname{Hol}(G)$: Every automorphism of $G$ extends to an inner automorphism in $\operatorname{Hol}(G)$.