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:

1. It is the subgroup generated by all its commutators
2. It is the normal closure of the subgroup generated by all its commutators
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 commutator subgroup or derived subgroup of a group $G$, denoted as $[G,G]$ or as $G^{\prime }$, is defined in the following way:

1. It is the subgroup generated by all commutators, or elements of the form $[x,y]=x^{-1}{y}^{-1}xy$
2. It is the normal closure of the subgroup generated by all its commutators
3. it is the intersection of all Abelian-quotient subgroups of $G$, viz., subgroups $H◃G$ such that $G/H$ is an Abelian group.

Equivalence of definitions

For full proof, refer: Equivalence of definitions of commutator subgroup

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 $n^{th}$ 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 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

is the original group and

dg

is the derived subgroup.

References

Textbook references

• Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632, ^{More info}, Page 234, Exercise 9 of Section 8 (Generators and relations) (definition introduced in exercise)
• Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, ^{More info}, Page 89
• Algebra by Serge Lang, ISBN 038795385X, ^{More info}, Page 20 (definition introduced in paragraph)
• Algebra (Graduate Texts in Mathematics) by Thomas W. Hungerford, ISBN 0387905189, ^{More info}, Page 102, Definition 7.7 (formal definition)
• A First Course in Abstract Algebra (6th Edition) by John B. Fraleigh, ISBN 0201763907, ^{More info}, Page 179, Exercise 33 (definition introduced in exercise)
• Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261, ^{More info}, Page 17, as derived subgroup (definition introduced in paragraph)