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).