# Normal subgroup contained in centralizer of derived subgroup

## Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: normal subgroup and subgroup contained in centralizer of commutator subgroup
View other subgroup property conjunctions | view all subgroup properties

## Definition

A subgroup $H$ of a group $G$ is termed a normal subgroup contained in centralizer of commutator subgroup if $H$ is a normal subgroup of $G$ and $H \le C_G([G,G])$, i.e., $H$ is contained in the centralizer of derived subgroup of $G$.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
normal subgroup whose automorphism group is abelian normal subgroup, also a group whose automorphism group is abelian (automorphism group is an abelian group) derived subgroup centralizes normal subgroup whose automorphism group is abelian
cyclic normal subgroup normal subgroup, also a cyclic group (via abelian automorphism group) (via abelian automorphism group) Normal subgroup whose automorphism group is abelian|FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
subgroup contained in centralizer of commutator subgroup
commutator-in-center subgroup commutator with whole group is contained in its center
commutator-in-centralizer subgroup
hereditarily 2-subnormal subgroup
class two normal subgroup

### Related group properties

A group $H$ has the property that for any group $G$ containing $H$ as a normal subgroup, $H$ is also contained in the centralizer of commutator subgroup of $G$, if and only if $H$ is a group whose automorphism group is abelian.