Self-centralizing normal subgroup

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

Relation with other properties

Stronger properties

Weaker properties

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