C-closed 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: c-closed 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 c-closed normal subgroup if it is both a c-closed subgroup (it equals the centralizer of its centralizer) and a normal subgroup (it is invariant under inner automorphisms).

Relation with other properties

Stronger properties

Metaproperties

Intersection-closedness

YES: This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property.
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ABOUT INTERSECTION-CLOSEDNESS: View all intersection-closed subgroup properties (or, strongly intersection-closed properties) | View all subgroup properties that are not intersection-closed | Read a survey article on proving intersection-closedness | Read a survey article on disproving intersection-closedness

An arbitrary intersection of c-closed normal subgroups is c-closed normal. This follows from the corresponding facts being true for c-closed subgroups and normal subgroups individually. For full proof, refer: c-closedness in strongly intersection-closed, normality is strongly intersection-closed

Centralizer-closedness

This subgroup property is centralizer-closed: the centralizer of any subgroup with this property, in the whole group, again has this property
View other centralizer-closed subgroup properties

The centralizer of a c-closed normal subgroup is again c-closed normal. This again follows from the corresponding statements for both properties. For full proof, refer: c-closedness is centralizer-closed, normality is centralizer-closed