Centralizer of derived subgroup

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

• 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 $[x_1,[x_2,x_3]]$ and the letter $x_1$. 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 $H$ 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