Derived subgroup: Difference between revisions

Latest revision as of 21:04, 26 December 2023

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 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 $G$ , denoted as $[G, G]$ or as $G^{'}$ , is defined in the following way:

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

It is a normal subgroup: derived subgroup is normal

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 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.

	Group part	Subgroup part	Quotient part
Center of dihedral group:D8	Dihedral group:D8	Cyclic group:Z2	Klein four-group

Examples where the derived subgroup is trivial

Further information: Derived subgroup is trivial if and only if group is abelian

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

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

is the original group and

dg

is the derived subgroup.

Warning: Commutator subgroup need not contain only commutators

Since the commutator subgroup of a group is the subgroup generated by the commutators, a priori, it need not consist only of commutators themselves. Indeed, groups whose commutator subgroup is a larger set than the set of commutators do exist. This mistake is sometimes made, since the smallest finite examples are large, obscure groups; the smallest finite groups with said property are of order 96. There are two of them, SmallGroup(96,3) and SmallGroup(96,203).

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)

@@ Line 1: / Line 1: @@
 {{subgroup-defining function}}
+[[Category: Derived subgroups]]
 ==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.
+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==
@@ Line 9: / Line 10: @@
 ===Symbol-free definition===
-The '''commutator subgroup''' or '''derived subgroup''' of a [[group]] is defined in the following equivalent ways:
+The '''derived subgroup''' or '''commutator subgroup''' of a [[group]] is defined in the following equivalent ways:
-# It is the subgroup generated by all its [[commutator]]s
+# It is the subgroup generated by all [[commutator]]s of the whole group.
-# It is the normal closure of the subgroup generated by all its commutators
+# It is the [[normal closure]] of the subgroup generated by all [[commutator]]s of the whole group.
-# It is the intersection of all [[Abelian-quotient subgroup]]s (viz., [[normal subgroup]]s with Abelian quotients). In other words, it is the smallest normal subgroup for which the quotient group is Abelian.
+# It is the intersection of all [[abelian-quotient subgroup]]s (viz., [[normal subgroup]]s 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]] <math>G</math>, denoted as <math>[G,G]</math> or as <math>G'</math>, is defined in the following way:
+The '''derived subgroup''' or '''commutator subgroup''' of a [[group]] <math>G</math>, denoted as <math>[G,G]</math> or as <math>\! G'</math>, is defined in the following way:
-# It is the subgroup generated by all commutators, or elements of the form <math>[x,y] = x^{-1}y^{-1}xy</math>
+# It is the subgroup generated by all [[commutator]]s, or elements of the form <math>[x,y] = xyx^{-1}y^{-1}</math> where <math>x,y \in G</math>.
-# It is the normal closure of the subgroup generated by all its commutators
+# It is the [[normal closure]] of the subgroup generated by all elements of the form <math>[x,y]</math>.
-# it is the intersection of all [[Abelian-quotient subgroup]]s of <math>G</math>, viz., subgroups <math>H \triangleleft G</math> such that <math>G/H</math> is an [[Abelian group]].
+# it is the intersection of all [[abelian-quotient subgroup]]s of <math>G</math>, viz., subgroups <math>H \underline{\triangleleft} G</math> such that <math>G/H</math> is an [[abelian group]].
 ===Equivalence of definitions===
-{{proofat|[[Equivalence of definitions of commutator subgroup]]}}
+{{proofat|[[Equivalence of definitions of derived subgroup]]}}
-==Group properties satisfied==
+==Group properties==
-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.
+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.
-==Subgroup properties satisfied==
+It is a normal subgroup: [[derived subgroup is normal]]
-* [[Verbal subgroup]]: In fact it is a [[verbal subgroup-defining function]] so it returns a verbal subgroup
+==Associated constructions==
-* [[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
+{{associated qdf|Abelianization}}
-* [[Upward-closed normal subgroup]]: Any subgroup containing the commutator subgroup is normal in the whole group. Hence the commutator subgroup is upward-closed normal.
+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 ds|Derived series}}
+The series obtained by iterating the commutator subgroup-defining function is termed the derived series. The <math>n^{th}</math> member of this is termed the <math>n^{th}</math> 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.
+{{sdf examples}}
+===Examples where the derived subgroup is trivial===
+{{further|[[Derived subgroup is trivial if and only if group is abelian]]}}
+These are precisely the [[abelian group]]s (follow through the link for examples).
+===Examples where the derived subgroup is the whole group===
+These are precisely the [[perfect group]]s (follow through the link for examples).
+==Subgroup properties==
+===Properties satisfied===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of satisfaction
+|-
+| [[Satisfies property::Verbal subgroup]] || generated by a bunch of words with letters freely quantified over the whole group || the word here is the [[commutator]]
+|-
+| [[Satisfies property::Fully invariant subgroup]] || invariant under all [[endomorphism]]s || [[derived subgroup is fully invariant]], see also [[verbal implies fully invariant]]
+|-
+| [[Satisfies property::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]]
+|-
+| [[Satisfies property::Characteristic subgroup]] || invariant under all [[automorphism]]s || [[derived subgroup is characteristic]], also follows from being a verbal subgroup
+|-
+| [[Satisfies property::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]]
+|-
+| [[Satisfies property::Normal subgroup]] || invariant under all [[inner automorphism]]s || (via characteristic)
+|-
+| [[Satisfies property::Abelian-quotient subgroup]] || quotient group is abelian ||
+|}
 ==Effect of operators==
@@ Line 53: / Line 100: @@
 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 qdf|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 ds|Derived series}}
-The series obtained by iterating the commutator subgroup-defining function is termed the derived series. The <math>n^{th}</math> member of this is termed the <math>n^{th}</math> 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==
@@ Line 83: / Line 119: @@
 We can assign this as a value, to a new name, for instance:
-<pre>dg = DerivedSubgroup (g);</pre>
+<pre>dg := DerivedSubgroup (g);</pre>
 where <pre>g</pre> is the original group and <pre>dg</pre> is the derived subgroup.
+==Warning: Commutator subgroup need not contain only commutators==
+Since the commutator subgroup of a group is the subgroup ''generated by'' the commutators, a priori, it need not consist only of commutators themselves. Indeed, groups whose commutator subgroup is a larger set than the set of commutators do exist. This mistake is sometimes made, since the smallest finite examples are large, obscure groups; the smallest finite groups with said property are of [[order 96]]. There are two of them, [[SmallGroup(96,3)]] and [[SmallGroup(96,203)]].
 ==References==