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
View other subgroup property conjunctions | view all subgroup properties

Definition

Symbol-free definition

A subgroup of a group is termed a self-centralizing normal subgroup or normal self-centralizing subgroup if it satisfies the following equivalent conditions:

• 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 $H$ of a group $G$ is termed a self-centralizing normal subgroup if it satisfies the following equivalent conditions:

• $H\triangleleft G$ and $C_{G}(H)\leq H$.
• The induced homomorphism $G/Z(H)\to \operatorname {Aut} (H)$, via the action by conjugation, is injective.

Relation with other properties

Stronger properties

• Critical subgroup
• Self-centralizing characteristic subgroup

Weaker 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

Metaproperties

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