Derived subgroup

History
The notion of derived subgroup or 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.

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


 * 1) It is the subgroup generated by all commutators of the whole group.
 * 2) It is the normal closure of the subgroup generated by all commutators of the whole group.
 * 3) It is the intersection of all abelian-quotient subgroups (viz., normal subgroups with abelian quotients). In other words, it is the smallest normal subgroup for which the quotient group is abelian.

Definition with symbols
The derived subgroup or commutator subgroup of a group $$G$$, denoted as $$[G,G]$$ or as $$\! G'$$, is defined in the following way:


 * 1) It is the subgroup generated by all commutators, or elements of the form $$[x,y] = xyx^{-1}y^{-1}$$ where $$x,y \in G$$.
 * 2) It is the normal closure of the subgroup generated by all elements of the form $$[x,y]$$.
 * 3) it is the intersection of all abelian-quotient subgroups of $$G$$, viz., subgroups $$H \underline{\triangleleft} G$$ such that $$G/H$$ is an abelian group.

Group properties
It is not true that every group can be realized as the derived subgroup of another group -- for instance, the characteristically metacyclic and commutator-realizable implies abelian tells us that a group whose first two abelianizations are cyclic, but whose second derived subgroup is not trivial, cannot arise as a derived subgroup.

Associated constructions
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.

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

Examples where the derived subgroup is proper and nontrivial
The quotient part in the examples below is the abelianization of the group.

Examples where the derived subgroup is trivial
These are precisely the abelian groups (follow through the link for examples).

Examples where the derived subgroup is the whole group
These are precisely the perfect groups (follow through the link for examples).

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

The 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.

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 is the original group and dg is the derived subgroup.

Textbook references

 * , Page 234, Exercise 9 of Section 8 (Generators and relations) (definition introduced in exercise)
 * , Page 89
 * , Page 20 (definition introduced in paragraph)
 * , Page 102, Definition 7.7 (formal definition)
 * , Page 179, Exercise 33 (definition introduced in exercise)
 * , Page 17, as derived subgroup (definition introduced in paragraph)