# 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 of a group is termed **fully normalized** if it satsifies the following equivalent conditions:

- For any automorphism of , there is an element in such that for all in .
- The map that sends to the automorphism is surjective (that is, its image is the whole of ).

## Relation with other properties

### Conjunction with other properties

- Normal fully normalized subgroup: This can be viewed as a subgroup where the map 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 is fully normalized in , and is a group containing , then is fully normalized in . 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.ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitiveABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

Since the property of being fully normalized is right-hereditary, it is automatically transitive.