# Centralizer of derived subgroup

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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

## 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 $G$ is defined as the subgroup $C_G([G,G])$.

## 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 $H$ such that $H/C_G([G,G]) = Z(G/C_G([G,G]))$.

## 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