# Commutator-in-center subgroup

From Groupprops

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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 commutator-in-centralizer subgroup

View other subgroup property conjunctions | view all subgroup properties

## Definition

A subgroup of a group is termed a **commutator-in-center** subgroup if it satisfies the following equivalent conditions:

- where denotes the commutator of two subgroups and denotes the center of .
- is a normal subgroup of and is trivial, i.e., is a commutator-in-centralizer subgroup.

A group has this property as a subgroup of itself if and only if it has nilpotency class two.

## Relation with other properties

### Stronger properties

### Weaker properties

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

Class two normal subgroup | normal subgroup of nilpotency class two | |FULL LIST, MORE INFO | ||

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

Hereditarily 2-subnormal subgroup | every subgroup is 2-subnormal in whole group | (via commutator-in-centralizer) | Commutator-in-centralizer subgroup|FULL LIST, MORE INFO | |

Normal subgroup | ||||

2-subnormal subgroup |

### Related group properties

A group has the property that for every group containing as a normal subgroup, is a commutator-in-center subgroup of , if and only if is a group whose inner automorphism group is central in automorphism group.