# Fully normalized subgroup

In

mathematics , in the field ofgroup theory , asubgroup of a group is said to be**fully normalized**if every automorphism of the subgroup lifts to aninner automorphism of the whole group. Another way of putting this is that the natural embedding from theWeyl group of the subgroup to itsautomorphism group is surjective.In symbols, a subgroup $H$ is fully normalized in $G$ if, given an automorphism $sigma$ of $H$, there is a $g\; in\; G$ such that the map $x\; mapsto\; gxg^\{-1\}$, when restricted to $H$ is equal to $sigma$.

Some facts:

* Every group can be embedded as a normal and fully normalized subgroup of a bigger group. A natural construction for this is the holomorph, which is itssemidirect product with its automorphism group.

* Acomplete group is fully normalized in any bigger group in which it is embedded because every automorphism of it is inner.

* Every fully normalized subgroup has theautomorphism extension property .

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