# Normal subgroup contained in centralizer of derived subgroup

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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 of a group is termed a **normal subgroup contained in centralizer of commutator subgroup** if is a normal subgroup of and , i.e., is contained in the centralizer of derived subgroup of .

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