# Centralizer of derived subgroup

From Groupprops

This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup

View a complete list of subgroup-defining functions OR View a complete list of quotient-defining functions

## Contents

## Definition

### Symbol-free definition

The **centralizer of derived subgroup** or **centralizer of commutator subgroup** of a group is defined as the centralizer of its derived subgroup.

### Definition with symbols

The **centralizer of commutator subgroup** or **centralizer of derived subgroup** of a group is defined as the subgroup .

## Facts

- Derived subgroup centralizes cyclic normal subgroup: In particular, this means that any cyclic normal subgroup is contained in the centralizer of derived subgroup.
- Derived subgroup centralizes aut-abelian normal subgroup: In particular, this means that any aut-abelian normal subgroup is contained in the centralizer of derived subgroup.
- Abelian-quotient abelian normal subgroup is contained in centralizer of derived subgroup
- Abelian subgroup is contained in centralizer of derived subgroup in generalized dihedral group
- Abelian subgroup equals centralizer of derived subgroup in generalized dihedral group unless it is a 2-group of exponent at most four

## Subgroup properties satisfied

- Weakly marginal subgroup: Corresponds to the word and the letter . See also centralizer of verbal subgroup is weakly marginal.
- Strictly characteristic subgroup
- Finite direct power-closed characteristic subgroup
- Characteristic subgroup

## Group properties satisfied

- Group of nilpotency class two: In fact, this subgroup is either abelian or has nilpotence class precisely two.

## Relation with other subgroup-defining functions

### Smaller subgroup-defining functions

Subgroup-defining function | Meaning |
---|---|

center | centralizes every element of the group |

second center | second member of upper central series |

### Larger subgroup-defining functions

Subgroup-defining function | Meaning |
---|---|

marginal subgroup for variety of metabelian groups | defined as the subgroup such that . |

## Effect of operators

### Fixed-point operator

A group equals its own centralizer of commutator subgroup if and only if it is nilpotent of class two.

## Subgroup-defining function properties

### Idempotence

This subgroup-defining function is idempotent. In other words, applying this twice to a given group has the same effect as applying it once