Self-centralizing normal subgroup
This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: self-centralizing subgroup and normal subgroup
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- It is normal and self-centralizing in the whole group (i.e., it contains its centralizer in the whole group).
- The induced map from the quotient of the whole group by its center, to the automorphism group of the subgroup, is injective.
Definition with symbols
A subgroup of a group is termed a self-centralizing normal subgroup if it satisfies the following equivalent conditions:
- and .
- The induced homomorphism , via the action by conjugation, is injective.
Relation with other properties
- Coprime automorphism-faithful subgroup (for finite groups): For full proof, refer: Self-centralizing and normal implies coprime automorphism-faithful
- Normality-large normal subgroup and Normality-large subgroup: For full proof, refer: Self-centralizing and normal implies normality-large
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