# Subgroup contained in centralizer of derived subgroup

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Definition

A subgroup of a group is termed a **subgroup contained in centralizer of commutator subgroup** if is contained in , the centralizer of commutator subgroup of .

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Central subgroup | contained in center | |FULL LIST, MORE INFO | ||

Cyclic normal subgroup | cyclic and a normal subgroup | Normal subgroup contained in centralizer of derived subgroup, Normal subgroup whose automorphism group is abelian|FULL LIST, MORE INFO | ||

Aut-abelian normal subgroup | normal subgroup whose automorphism group is abelian | aut-abelian normal subgroup is contained in centralizer of commutator subgroup | Normal subgroup contained in centralizer of derived subgroup|FULL LIST, MORE INFO | |

Normal subgroup contained in centralizer of commutator subgroup | normal and contained in centralizer of commutator subgroup |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Hereditarily 2-subnormal subgroup | every subgroup is 2-subnormal | centralizer of commutator subgroup is hereditarily 2-subnormal | Commutator-in-centralizer subgroup|FULL LIST, MORE INFO | |

Commutator-in-centralizer subgroup | commutator with whole group is contained in centralizer | |||

2-subnormal subgroup | normal subgroup of normal subgroup | Commutator-in-centralizer subgroup|FULL LIST, MORE INFO |