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 subgroupProperty "Conjunction involving" (as page type) with input value "</br>self-centralizing subgroup" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.Property "Defining ingredient" (as page type) with input value "</br>self-centralizing subgroup" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process. and

normal subgroupProperty "Conjunction involving" (as page type) with input value "</br>normal subgroup" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.Property "Defining ingredient" (as page type) with input value "</br>normal subgroup" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.
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Definition

Symbol-free definition

A subgroup of a group is termed a self-centralizing normal subgroup or normal self-centralizing subgroup if it is normal and self-centralizing in the whole group (i.e., it contains its centralizer in the whole group).

Definition with symbols

A subgroup $H$ of a group $G$ is termed a self-centralizing normal subgroup if $H\triangleleft G$ and $C_{G}(H)\leq H$.

Relation with other properties

Stronger properties

• Critical subgroup
• Self-centralizing characteristic subgroup

Weaker properties

• Coprime automorphism-faithful subgroup (for finite groups)
• 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