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]).

Facts

Subgroup properties satisfied

Group properties satisfied

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