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
- 1 Definition
- 2 Facts
- 3 Subgroup properties satisfied
- 4 Group properties satisfied
- 5 Relation with other subgroup-defining functions
- 6 Effect of operators
- 7 Subgroup-defining function properties
Definition with symbols
The centralizer of commutator subgroup or centralizer of derived subgroup of a group is defined as the subgroup .
- 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
|center||centralizes every element of the group|
|second center||second member of upper central series|
Larger subgroup-defining functions
|marginal subgroup for variety of metabelian groups||defined as the subgroup such that .|
Effect of operators
A group equals its own centralizer of commutator subgroup if and only if it is nilpotent of class two.
Subgroup-defining function properties
This subgroup-defining function is idempotent. In other words, applying this twice to a given group has the same effect as applying it once