Central factor of normalizer
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]
This is a variation of central factor|Find other variations of central factor |
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A subgroup of a group is termed a central factor of normalizer 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 of a group is termed a central factor of normalizer if it satisfies the following equivalent conditions:
- where denotes the centralizer of in .
- For any in such that , there exists an element in such that for any in , .
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 of a group 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 and outputs the property of being a subgroup that satisfies property 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
- 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
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: View variations of this property that are transitive|View variations of this property that are not transitive
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.
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: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition
If and be a WC-subgroup of . Then, any inner automorphism of arises from an inner automorphism of , hence the restriction of an inner automorphism of to is also the restriction of an inner automorphism of to . Hence, it must be an inner automorphism of .
Effect of property operators
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.
The property of being a WC subgroup is not transitive. Thus, there is a notion of sub-WC-subgroup.