Fully normalized subgroup

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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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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

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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


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.