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

History

The notion of commutator subgroup naturally arose in the context of finding a natural choice for a good composition series for a solvable group -- solvable groups arise very naturally in the fundamental problems dealt with in Galois theory.

Definition

Symbol-free definition

The commutator subgroup or derived subgroup of a group is defined in the following equivalent ways:

It is the subgroup generated by all its commutators
It is the intersection of all Abelian-quotient subgroups (viz normal subgroups with Abelian quotients)

Definition with symbols

The commutator subgroup or derived subgroup of a group $G$ , denoted as $[G, G]$ or as $G^{'}$ , is defined as the subgroup generated by all commutators, or elements of the form $[x, y] = x^{- 1} y^{- 1} x y$ .

Group properties satisfied

It is not true that every group can be realized as the commutator subgroup of another group == for instance, the cyclic Abelianizations theorem tells us that a group whose first two Abelianizations are cyclic, but whose second derived subgroup is not trivial, cannot arise as a commutator subgroup.

Subgroup properties satisfied

Verbal subgroup: In fact it is a verbal subgroup-defining function so it returns a verbal subgroup
Fully characteristic subgroup: This follows from the fact that every verbal subgroup is fully characteristic
Characteristic subgroup: This follows from the fact that every verbal subgroup is characteristic
Upward-closed normal subgroup: Any subgroup containing the commutator subgroup is normal in the whole group. Hence the commutator subgroup is upward-closed normal.

Effect of operators

Fixed-point operator

A group which equals its own commutator subgroup is termed a perfect group

Free operator

A group whose commutator subgroup is trivial is termed an Abelian group

Subgroup-defining function properties

Monotonicity

This subgroup-defining function is monotone, viz the image of any subgroup under this function is contained in the image of the whole group

This follows from the fact that any commutator of elements inside a subgroup is also a commutator of elements inside the whole group.

Associated constructions

Associated quotient-defining function

The quotient-defining function associated with this subgroup-defining function is: [[Abelianization]]

The quotient of a group by its commutator subgroup is termed its Abelianization. This can also be thought of as the largest possible Abelian quotient of the group.

Associated descending series

The associated descending series to this subgroup-defining function is: [[Derived series]]

The series obtained by iterating the commutator subgroup-defining function is termed the derived series. The $n^{t h}$ member of this is termed the $n^{t h}$ derived subgroup.

A group for which this derived series terminates at the identity in finitely many steps is termed a solvable group and the length of the derived series is termed the solvable length.

Computation

The computation problem

Further information: Commutator subgroup computation problem

The general problem of computing the commutator subgroup given the whole group can be solved, when ther group is described in terms of a generating set. The idea is to take the normal closure of the subgroup generated by all commutators of elements in the generating set.

GAP command

The command for computing this subgroup-defining function in Groups, Algorithms and Programming (GAP) is:DerivedSubgroup
View other GAP-computable subgroup-defining functions

To compute the commutator subgroup of a group in GAP, the syntax is:

DerivedSubgroup (group);

where

group

could either be an on-the-spot description of the group or a name aluding to a previously defined group.

We can assign this as a value, to a new name, for instance:

dg = DerivedSubgroup (g);

where

g</math> is the original group and  $d g$  is the derived subgroup.