# Commutator-in-centralizer subgroup

From Groupprops

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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 **commutator-in-centralizer subgroup** if the commutator is contained in the centralizer , or equivalently, is trivial.,

## Relation with other properties

### Stronger properties

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

Commutator-in-center subgroup | commutator with whole group is contained in center of subgroup | |FULL LIST, MORE INFO | ||

Abelian normal subgroup | Commutator-in-center subgroup|FULL LIST, MORE INFO | |||

Subgroup contained in centralizer of commutator subgroup | contained in the centralizer of the whole group's commutator subgroup | |FULL LIST, MORE INFO | ||

Aut-abelian normal subgroup | Commutator-in-center subgroup, Normal subgroup contained in centralizer of derived subgroup, Subgroup contained in centralizer of derived subgroup|FULL LIST, MORE INFO |

### Weaker properties

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

Hereditarily 2-subnormal subgroup | ||||

Class two 2-subnormal subgroup | ||||

2-subnormal subgroup |