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
Contents
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.
Definition
Symbol-free definition
The derived subgroup or commutator subgroup of a group is defined in the following equivalent ways:
- It is the subgroup generated by all commutators of the whole group.
- It is the normal closure of the subgroup generated by all commutators of the whole group.
- 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 , denoted as or as , is defined in the following way:
- It is the subgroup generated by all commutators, or elements of the form where .
- It is the normal closure of the subgroup generated by all elements of the form .
- it is the intersection of all abelian-quotient subgroups of , viz., subgroups such that is an abelian group.
Equivalence of definitions
For full proof, refer: Equivalence of definitions of derived subgroup
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
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 member of this is termed the 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
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).
Subgroup properties
Properties satisfied
Property | Meaning | Proof of satisfaction |
---|---|---|
Verbal subgroup | generated by a bunch of words with letters freely quantified over the whole group | the word here is the commutator |
Fully invariant subgroup | invariant under all endomorphisms | derived subgroup is fully invariant, see also verbal implies fully invariant |
Image-closed fully invariant subgroup | under any surjective homomorphism, its image is fully invariant in the image of the whole group | follows from verbal implies image-closed fully invariant |
Characteristic subgroup | invariant under all automorphisms | derived subgroup is characteristic, also follows from being a verbal subgroup |
Upward-closed normal subgroup | any subgroup containing it is a normal subgroup | derived subgroup is upward-closed normal -- follows from the fact that the quotient is abelian and abelian implies every subgroup is normal |
Normal subgroup | invariant under all inner automorphisms | (via characteristic) |
Abelian-quotient subgroup | quotient group is abelian |
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.
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
groupcould 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
gis the original group and
dgis 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)