Groupprops, The Group Properties Wiki (pre-alpha)

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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.
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VIEW RELATED: Subgroup property implications |
VIEW FACTS THAT:| prove, or show that, a subgroup satisfies the property
RANDOM SUBGROUP PROPERTY: Conjugacy-closed subgroup: A subgroup such that any two elements of the subgroup that are conjugate in the whole group are conjugate in the subgroup.

This is a variation of central factor
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Contents 1 Definition 1.1 Symbol-free definition 1.2 Definition with symbols 2 Formalisms 2.1 First-order description 2.2 In terms of the in-normalizer operator 2.3 In terms of the when-defined restriction formalism 3 Relation with other properties 3.1 Stronger properties 3.2 Weaker properties 4 Metaproperties 4.1 Transitivity 4.2 Intersection-closedness 4.3 Intermediate subgroup condition 5 Effect of property operators 5.1 Right transiter 5.2 Subordination

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Definition

Symbol-free definition

A subgroup of a group is termed a WC-subgroup if it satisfies the following equivalent conditinos:

• It is a central factor of its normalizer
• Every inner automorphism of the group that restricts to an automorphism on the subgroup restricts to an inner automorphism on the subgroup.

Definition with symbols

A subgroup H of a group G is termed a WC-subgroup if it satisfies the following equivalent conditions:

• HC_G(H) = N_G(H) where C_G(H) denotes the centralizer of H in G.
• For any g in G such that gHg ^{− 1} = H, there exists an element h in H such that for any x in G, gxg ^{− 1} = hxh ^{− 1}.

Formalisms

First-order description

This subgroup property is a first-order subgroup property, viz., it has a first-order description in the theory of groups.
View a complete list of first-order subgroup properties

A subgroup H of a group G is a WC-subgroup if it satisfies the following condition:

In terms of the in-normalizer operator

This property is obtained by applying the in-normalizer operator to the property: central factor
View other properties obtained by applying the in-normalizer operator

A subgroup is a WC-subgroup if and only if it is a central factor of its normalizer. This can be expressed in terms of thein-normalizer operator. The in-normalizer operator takes a subgroup property p and outputs the property of being a subgroup that satisfies property p inside the normalizer.

In terms of the when-defined restriction formalism

The property of being a WC subgroup is the balanced property with respect to the when-defined function restriction formalism. Note that balanced properties for when-defined formalisms are not necessarily transitive.

Relation with other properties

Stronger properties

• Central factor: Its normalizer is the whole group, and it is clearly a central factor in that
• Self-normalizing subgroup: Its normalizer is itself, and it is clearly a central factor of itself.
• Cocentral subgroup

Weaker properties

• Sub-WC-subgroup

Metaproperties

Transitivity

NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole group
ABOUT THIS PROPERTY: |
ABOUT TRANSITIVITY: View a complete list of subgroup properties that are not transitive| View facts related to transitivity of subgroup properties | View a survey article on disproving transitivity

Check out the property operators part of this page for more details.

Intersection-closedness

It seems unlikely that an intersection of WC-subgroups should again be a WC-subgroup.

Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: |
ABOUT INTERMEDIATE SUBROUP CONDITION: View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

If and G be a WC-subgroup of K. Then, any inner automorphism of H arises from an inner automorphism of K, hence the restriction of an inner automorphism of H to G is also the restriction of an inner automorphism of K to G. Hence, it must be an inner automorphism of G.

Effect of property operators

Right transiter

Any WC-subgroup of a central factor is a WC-subgroup. Thus, the right transiter of the property of being a WC-subgroup is weaker than the property of being a central factor. It's not clear whether it is exactly equal.

Subordination

The property of being a WC subgroup is not transitive. Thus, there is a notion of sub-WC-subgroup.