Fully normalized subgroup
This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.
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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.
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.
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.
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.
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.