Centralizer of derived subgroup

Symbol-free definition
The centralizer of derived subgroup or centralizer of commutator subgroup of a group is defined as the defining ingredient::centralizer of its defining ingredient::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.

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