Fully normalized subgroup

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Definition

Symbol-free definition

A subgroup of a group is termed fully normalized if it satisfies the following equivalent conditions:

• Every automorphism of the subgroup lifts to an inner automorphism of the whole group
• The Weyl group (viz the image of the canonical map from the normalizer of the subgroup to its automorphism group), is the whole automorphism group

Definition with symbols

A subgroup $H$ of a group $G$ is termed fully normalized if it satsifies the following equivalent conditions:

• For any automorphism $\sigma$ of $H$, there is an element $g$ in $G$ such that $\sigma(x) = gxg^{-1}$ for all $g$ in $G$.
• The map $c: N_G(H) \to Aut(H)$ that sends $g \in N_G(H)$ to the automorphism $h \mapsto ghg^{-1}$ is surjective (that is, its image is the whole of $Aut(H)$).

Relation with other properties

Conjunction with other properties

• Normal fully normalized subgroup: This can be viewed as a subgroup where the map $c:G \to Aut(H)$ given by conjugation is well-defined and surjective.

Metaproperties

Trimness

The subgroup property of being fully normalized is trivially true, that is, the trivial subgroup is fully normalized in any group. It is not identity-true. In fact, a group satisfies the property as a subgroup of itself if and only if every automorphism of the group is inner.

Left-realized

Every group can be embedded as a fully normalized subgroup of some group (in fact, as a normal fuly normalized subgroup of itself). A natural example of such a group is the holomorph of the given group, which is the semidirect product of the group with its automorphism group. We say that the property of being fully normalized is a left-realized subgroup property.

Right-hereditariness

This subgroup property is right-hereditary: if a subgroup has the property in a group, it has the property in every bigger group. Hence, it is also a transitive subgroup property.

If $H$ is fully normalized in $K$, and $G$ is a group containing $K$, then $H$ is fully normalized in $G$. Thus, the property of being fully normalized is a right-hereditary subgroup property.

Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
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Since the property of being fully normalized is right-hereditary, it is automatically transitive.