Difference between revisions of "Abelian group"

From Groupprops
Jump to: navigation, search
(Relation with other properties)
(Weaker properties)
 
(19 intermediate revisions by the same user not shown)
Line 33: Line 33:
 
===Equivalent formulations===
 
===Equivalent formulations===
  
* A group is abelian if its [[defining ingredient::center]] is the whole group.
+
A group <math>G</math> is termed abelian if it satisfies the following equivalent conditions:
* A group is abelian if its [[defining ingredient::derived subgroup]] is trivial.
+
 
 +
* Its [[defining ingredient::center]] <math>Z(G)</math> is the whole group.
 +
* Its [[defining ingredient::derived subgroup]] <math>G' = [G,G]</math> is trivial.
 +
* (Choose a generating set <math>S</math> for <math>G</math>). For any elements <math>a,b \in S</math>, <math>ab = ba</math>.
 +
* The diagonal subgroup <math>\{ (g,g) \mid g \in G \}</math> is a [[defining ingredient::normal subgroup]] inside <math>G \times G</math>.
 
<section begin=beginner/>
 
<section begin=beginner/>
  
Line 68: Line 72:
 
===Non-examples===
 
===Non-examples===
  
Not every group is abelian. The smallest non-abelian group is [[symmetric group:S3|the symmetric group on three letters]]: the group of all permutations on three letters, under composition. Its being non-abelian hinges on the fact that the order in which permutations are performed, matters.
+
Not every group is abelian. The smallest non-abelian group is [[symmetric group:S3|the symmetric group on three letters]]: the group of all permutations on three letters, under composition. Its being non-abelian hinges on the fact that the order in which permutations are performed matters.
 
<section end=revisit/>
 
<section end=revisit/>
  
Line 79: Line 83:
 
===Occurrence as quotients===
 
===Occurrence as quotients===
  
The maximal abelian quotient of any group is termed its [[abelianization]], and this is the quotient by the [[commutator subgroup]]. A subgroup is an [[abelian-quotient subgroup]] (i.e., normal with abelian quotient group) if and only if the subgroup contains the commutator subgroup.
+
The maximal abelian quotient of any group is termed its [[abelianization]], and this is the quotient by the [[derived subgroup]]. A subgroup is an [[abelian-quotient subgroup]] (i.e., normal with abelian quotient group) if and only if the subgroup contains the derived subgroup.
  
==Formalisms==
+
==Metaproperties==
  
{{obtainedbyapplyingthe|diagonal-in-square operator|normal subgroup}}
+
{| class="sortable" border="1"
 
+
! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
A group <math>G</math> is an abelian group if and only if, in the [[external direct product]] <math>G \times G</math>, the diagonal subgroup <math>\{ (g,g) \mid g \in G \}</math> is a [[normal subgroup]].
+
|-
 +
| [[satisfies metaproperty::varietal group property]] || Yes || || The collection of abelian groups forms a subvariety of the variety of groups. In particular, it is closed under taking subgroups, quotients, and arbitrary direct products
 +
|-
 +
| [[satisfies metaproperty::subgroup-closed group property]] || Yes || [[abelianness is subgroup-closed]] || If <math>G</math> is an abelian group and <math>H</math> is a subgroup of <math>G</math>, then <math>H</math> is abelian.
 +
|-
 +
| [[satisfies metaproperty::quotient-closed group property]] || Yes || [[abelianness is quotient-closed]] || If <math>G</math> is an abelian group and <math>H</math> is a normal subgroup of <math>G</math>, the [[quotient group]] <math>G/H</math> is abelian.
 +
|-
 +
| [[satisfies metaproperty::direct product-closed group property]] || Yes || [[abelianness is direct product-closed]] || Suppose <math>G_i, i \in I</math>, are abelian groups. Then, the external direct product <math>\prod_{i \in I} G_i</math> is also abelian.
 +
|}
  
 
==Relation with other properties==
 
==Relation with other properties==
Line 92: Line 104:
  
 
{| class="sortable" border="1"
 
{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! pProof of strictness (reverse implication failure) !! Intermediate notions !! Comparison
+
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions !! Comparison
 
|-
 
|-
| [[weaker than::cyclic group]] || generated by one element || [[cyclic implies abelian]] || [[abelian not implies cyclic]] {{strictness examples|abelian group|cyclic group}} || {{intermediate notions|abelian group|cyclic group}} ||
+
| [[weaker than::cyclic group]] || generated by one element || [[cyclic implies abelian]] || [[abelian not implies cyclic]] {{strictness examples|abelian group|cyclic group}} || {{intermediate notions short|abelian group|cyclic group}} ||
 
|-
 
|-
| [[weaker than::homocyclic group]] || direct product of isomorphic cyclic groups || || {{strictness examples|abelian group|homocyclic group}} || ||
+
| [[weaker than::homocyclic group]] || direct product of isomorphic cyclic groups || || {{strictness examples|abelian group|homocyclic group}} || {{intermediate notions short|abelian group|homocyclic group}}||
 
|-
 
|-
| [[weaker than::finite abelian group]] || abelian and a [[finite group]] || || {{strictness examples|abelian group|finite group}} ||
+
| [[Weaker than::residually cyclic group]] || every non-identity element is outside a normal subgroup with a cyclic quotient group || || {{strictness examples|abelian group|residually cyclic group}} || {{intermediate notions short|abelian group|residually cyclic group}} ||  
 
|-
 
|-
| [[weaker than::finitely generated abelian group]] || abelian and a [[finitely generated group]] || || {{strictness examples|abelian group|finitely generated group}} || ||
+
| [[Weaker than::locally cyclic group]] || every finitely generated subgroup is cyclic || || {{strictness examples|abelian group|locally cyclic group}} || {{intermediate notions short|abelian group|locally cyclic group}} ||
 +
|-
 +
| [[Weaker than::epabelian group]] || abelian group whose exterior square is the trivial group || || {{strictness examples|abelian group|epabelian group}} || {{intermediate notions short|abelian group|epabelian group}} ||
 +
|-
 +
| [[weaker than::finite abelian group]] || abelian and a [[finite group]] || || {{strictness examples|abelian group|finite group}} || {{intermediate notions short|abelian group|finite abelian group}}||
 +
|-
 +
| [[weaker than::finitely generated abelian group]] || abelian and a [[finitely generated group]] || || {{strictness examples|abelian group|finitely generated group}} || {{intermediate notions short|abelian group|finitely generated abelian group}}||
 
|}
 
|}
  
 
===Weaker properties===
 
===Weaker properties===
 
{| class="sortable" border="1"
 
{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! pProof of strictness (reverse implication failure) !! Intermediate notions !! Comparison
+
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions  
 +
|-
 +
| [[stronger than::nilpotent group]] || [[lower central series]] reaches identity, [[upper central series]] reaches whole group || [[abelian implies nilpotent]] || [[nilpotent not implies abelian]] {{strictness examples|nilpotent group|abelian group}} || {{intermediate notions short|nilpotent group|abelian group}}
 +
|-
 +
| [[stronger than::solvable group]] || [[derived series]] reaches identity, has [[normal series]] with abelian factor groups || [[abelian implies solvable]] || [[solvable not implies abelian]] {{strictness examples|solvable group|abelian group}} || {{intermediate notions short|solvable group|abelian group}}
 
|-
 
|-
| [[stronger than::Nilpotent group]] || [[lower central series]] reaches identity, [[upper central series]] reaches whole group || [[abelian implies nilpotent]] || [[nilpotent not implies abelian]] {{strictness examples|nilpotent group|abelian group}} || {{intermediate notions|nilpotent group|abelian group}} ||
+
| [[stronger than::metabelian group]] || has [[abelian normal subgroup]] with abelian quotient group || || {{strictness examples|metabelian group|abelian group}} || {{intermediate notions short|metabelian group|abelian group}}
 
|-
 
|-
| [[stronger than::Solvable group]] || [[derived series]] reaches identity, has [[normal series]] with abelian factor groups || [[abelian implies solvable]] || [[solvable not implies abelian]] {{strictness examples|solvable group|abelian group}} || {{intermediate notions|solvable group|abelian group}} ||
+
| [[stronger than::virtually abelian group]] || has abelian subgroup of finite index || || {{strictness examples|virtually abelian group|abelian group}} || {{intermediate notions short|virtually abelian group|abelian group}}
 
|-
 
|-
| [[stronger than::Metabelian group]] || has [[abelian normal subgroup]] with abelian quotient group || || {{strictness examples|metabelian group|abelian group}} ||
+
| [[stronger than::FZ-group]] || center has finite index || || {{strictness examples|FZ-group|abelian group}} || {{intermediate notions short|FZ-group|abelian group}}
 
|-
 
|-
| [[stronger than::Virtually abelian group]] || has abelian subgroup of finite index || || {{strictness examples|virtually abelian group|abelian group}} ||
+
| [[stronger than::FC-group]] || every conjugacy class is finite || || {{strictness examples|FC-group|abelian group}} || {{intermediate notions short|FC-group|abelian group}}
 
|}
 
|}
  
==Metaproperties==
+
===Incomparable properties===
 
 
{{varietal}}
 
  
Abelian groups form a [[variety of algebras]]. The defining equations for this variety are the equations for a [[group]] along with the commutativity equation.
+
* [[Supersolvable group]] is a group that has a [[normal series]] where all the successive quotient groups are [[cyclic group]]s. An abelian group is supersolvable if and only if it is [[finitely generated abelian group|finitely generated]].
 +
* [[Polycyclic group]] is a group that has a [[subnormal series]] where all the successive quotent groups are [[cyclic group]]s. An abelian group is polycyclic if and only if it is finitely generated.
  
{{S-closed}}
+
==Formalisms==
  
Any [[subgroup]] of an abelian group is abelian -- viz., the property of being abelian is [[subgroup-closed group property|subgroup-closed]]. This follows as a direct consequence of abelianness being varietal. {{proofat|[[Abelianness is subgroup-closed]]}}
+
{{obtainedbyapplyingthe|diagonal-in-square operator|normal subgroup}}
  
{{Q-closed}}
+
A group <math>G</math> is an abelian group if and only if, in the [[external direct product]] <math>G \times G</math>, the diagonal subgroup <math>\{ (g,g) \mid g \in G \}</math> is a [[normal subgroup]].
 
 
Any [[quotient]] of an abelian group is abelian -- viz the property of being abelian is [[quotient-closed group property|quotient-closed]]. This again follows as a direct consequence of abelianness being varietal. {{proofat|[[Abelianness is quotient-closed]]}}
 
 
 
{{DP-closed}}
 
 
 
A [[direct product]] of abelian groups is abelian -- viz the property of being abelian is [[direct product-closed group property|direct product-closed]]. This again follows as a direct consequence of abelianness being varietal. {{proofat|[[Abelianness is direct product-closed]]}}
 
  
 
==Testing==
 
==Testing==
Line 139: Line 154:
  
 
{{further|[[Abelianness testing problem]]}}
 
{{further|[[Abelianness testing problem]]}}
 +
 +
The abelianness testing problem is the problem of testing whether a group (described using some [[group description rule]], such as an [[encoding of a group]] or a [[multi-encoding of a group]]) is abelian.
 +
 +
Algorithms for the abelianness testing problem range from the [[brute-force black-box group algorithm for abelianness testing]] (that involves testing for ''every'' pair of elements whether they commute, and is quadratic in the order of the group) to the [[generating set-based black-box group algorithm for abelianness testing]] (that involves testing only on a generating set, and is quadratic in the size of the generating set).
  
 
{{GAP command for gp|
 
{{GAP command for gp|
Line 146: Line 165:
 
To test whether a group is abelian, the GAP syntax is:
 
To test whether a group is abelian, the GAP syntax is:
  
<pre>IsAbelian (group)</pre>
+
<tt>IsAbelian (group)</tt>
  
where <pre>group</pre> either defines the group or gives the name to a group previously defined.
+
where <tt>group</tt> either defines the group or gives the name to a group previously defined.
  
 
==Study of this notion==
 
==Study of this notion==
Line 156: Line 175:
 
==References==
 
==References==
 
===Textbook references===
 
===Textbook references===
* {{booklink-defined|DummitFoote}}, Page 17 (definition as Point (2) in general definition of a group)
+
 
* {{booklink-defined|AlperinBell}}, Page 2 (definition introduced in paragraph)
+
{| class="sortable" border="1"
* {{booklink-defined|Artin}}, Page 42 (defined immediately after the definition of group, as a group where the composition is commutative)
+
! Book !! Page number !! Chapter and section !! Contextual information !! View
* {{booklink-defined|Herstein}}, Page 28 (formal definition)
+
|-
* {{booklink-defined|RobinsonGT}}, Page 2 (formal definition)
+
| {{booklink-defined-tabular|DummitFoote|17|Formal definition (definition as point (2) in general definition of group)|}} ||
* {{booklink-defined|FGTAsch}}, Page 1 (definition introduced in paragraph)
+
|-
 +
| {{booklink-defined-tabular|AlperinBell|2|1.1 (Rudiments of Group Theory/Review)|definition introduced in paragraph}} || [https://books.google.com/books?id=EroGCAAAQBAJ&pg=PA2 Google Books]
 +
|-
 +
| {{booklink-defined-tabular|Artin|42||definition introduced in paragraph (immediately after definition of group)}} ||
 +
|-
 +
| {{booklink-defined-tabular|Herstein|28||Formal definition}} ||
 +
|-
 +
| {{booklink-defined-tabular|RobinsonGT|2|1.1 (Binary Operations, Semigroups, and Groups)|formal definition}} || [https://books.google.com/books?id=EroGCAAAQBAJ&pg=PA2 Google Books]
 +
|-
 +
| {{booklink-defined-tabular|FGTAsch|1|1.1 (Elementary group theory)|definition introduced in paragraph}} || [https://books.google.com/books?id=BprbtnlI6HEC&pg=PA1 Google Books]
 +
|}
  
 
==External links==
 
==External links==

Latest revision as of 15:35, 11 April 2017

This article is about a basic definition in group theory. The article text may, however, contain advanced material.
VIEW: Definitions built on this | Facts about this: (facts closely related to Abelian group, all facts related to Abelian group) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |[SHOW MORE]
This article defines a group property that is pivotal (i.e., important) among existing group properties
View a list of pivotal group properties | View a complete list of group properties [SHOW MORE]

History

Origin of the term

The term abelian group comes from Niels Henrick Abel, a mathematician who worked with groups even before the formal theory was laid down, in order to prove unsolvability of the quintic.

The word abelian is usually begun with a small a.

wikinote: Some older content on the wiki uses capital A for Abelian. We're trying to update this content.


Definition

An abelian group is a group where any two elements commute. In symbols, a group G is termed abelian if for any elements x and y in G, xy = yx (here xy denotes the product of x and y in G). Note that x,y are allowed to be equal, though equal elements commute anyway, so we can restrict attention if we wish to unequal elements.


Full definition

An abelian group is a set G equipped with a (infix) binary operation + (called the addition or group operation), an identity element 0 and a (prefix) unary operation -, called the inverse map or negation map, satisfying the following:

Equivalent formulations

A group G is termed abelian if it satisfies the following equivalent conditions:


Notation

When G is an abelian group, we typically use additive notation and terminology. Thus, the group multiplication is termed addition and the product of two elements is termed the sum.

  1. The infix operator + is used for the group multiplication, so the sum of two elements a and b is denoted by a + b. The group multiplication is termed addition and the product of two elements is termed the sum.
  2. The identity element is typically denoted as 0 and termed zero
  3. The inverse of an element is termed its negative or additive inverse. The inverse of a is denoted -a
  4. a + a + \ldots + a done n times is denoted na, (where n \in \mathbb{N}) while (-a) + (-a) + (-a) + \ldots + (-a) done n times is denoted (-n)a.

This convention is typically followed in a situation where we are dealing with the abelian group G in isolation, rather than as a subgroup of a possibly non-abelian group. If we are working with subgroups in a non-abelian group, we typically use multiplicative notation even if the subgroup happens to be abelian.

Examples

VIEW: groups satisfying this property | groups dissatisfying this property
VIEW: Related group property satisfactions | Related group property dissatisfactions

Some infinite examples

The additive group of integers \mathbb{Z}, the additive group of rational numbers \mathbb{Q}, the additive group of real numbers \mathbb{R}, the multiplicative group of nonzero rationals \mathbb{Q}^*, and the multiplicative group of nonzero real numbers \mathbb{R}^* are some examples of Abelian groups.

(More generally, for any field, the additive group, and the multiplicative group of nonzero elements, are Abelian groups).


Finite examples

Cyclic groups are good examples of abelian groups, where the cyclic group of order n is the group of integers modulo n.

Further, any direct product of cyclic groups is also an abelian group. Further, every finitely generated Abelian group is obtained this way. This is the famous structure theorem for finitely generated abelian groups.

The structure theorem can be used to generate a complete listing of finite abelian groups, as described here: classification of finite Abelian groups.

Non-examples

Not every group is abelian. The smallest non-abelian group is the symmetric group on three letters: the group of all permutations on three letters, under composition. Its being non-abelian hinges on the fact that the order in which permutations are performed matters.


Facts

Occurrence as subgroups

Every cyclic group is abelian. Since each group is generated by its cyclic subgroups, every group is generated by a family of abelian subgroups. A trickier question is: do there exist abelian normal subgroups? A good candidate for an abelian normal subgroup is the center, which is the collection of elements of the group that commute with every element of the group.

Occurrence as quotients

The maximal abelian quotient of any group is termed its abelianization, and this is the quotient by the derived subgroup. A subgroup is an abelian-quotient subgroup (i.e., normal with abelian quotient group) if and only if the subgroup contains the derived subgroup.

Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
varietal group property Yes The collection of abelian groups forms a subvariety of the variety of groups. In particular, it is closed under taking subgroups, quotients, and arbitrary direct products
subgroup-closed group property Yes abelianness is subgroup-closed If G is an abelian group and H is a subgroup of G, then H is abelian.
quotient-closed group property Yes abelianness is quotient-closed If G is an abelian group and H is a normal subgroup of G, the quotient group G/H is abelian.
direct product-closed group property Yes abelianness is direct product-closed Suppose G_i, i \in I, are abelian groups. Then, the external direct product \prod_{i \in I} G_i is also abelian.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions Comparison
cyclic group generated by one element cyclic implies abelian abelian not implies cyclic (see also list of examples) Epabelian group, Locally cyclic group, Residually cyclic group|FULL LIST, MORE INFO
homocyclic group direct product of isomorphic cyclic groups (see also list of examples) |FULL LIST, MORE INFO
residually cyclic group every non-identity element is outside a normal subgroup with a cyclic quotient group (see also list of examples) |FULL LIST, MORE INFO
locally cyclic group every finitely generated subgroup is cyclic (see also list of examples) Epabelian group|FULL LIST, MORE INFO
epabelian group abelian group whose exterior square is the trivial group (see also list of examples) |FULL LIST, MORE INFO
finite abelian group abelian and a finite group (see also list of examples) |FULL LIST, MORE INFO
finitely generated abelian group abelian and a finitely generated group (see also list of examples) Residually cyclic group|FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
nilpotent group lower central series reaches identity, upper central series reaches whole group abelian implies nilpotent nilpotent not implies abelian (see also list of examples) Group in which class equals maximum subnormal depth, Group of nilpotency class three, Group of nilpotency class two, Group of nilpotency class two whose commutator map is the double of an alternating bihomomorphism giving class two, UL-equivalent group|FULL LIST, MORE INFO
solvable group derived series reaches identity, has normal series with abelian factor groups abelian implies solvable solvable not implies abelian (see also list of examples) Metabelian group, Metanilpotent group, Nilpotent group|FULL LIST, MORE INFO
metabelian group has abelian normal subgroup with abelian quotient group (see also list of examples) Group of nilpotency class two|FULL LIST, MORE INFO
virtually abelian group has abelian subgroup of finite index (see also list of examples) FZ-group|FULL LIST, MORE INFO
FZ-group center has finite index (see also list of examples) |FULL LIST, MORE INFO
FC-group every conjugacy class is finite (see also list of examples) FZ-group, Group with finite derived subgroup|FULL LIST, MORE INFO

Incomparable properties

Formalisms

In terms of the diagonal-in-square operator

This property is obtained by applying the diagonal-in-square operator to the property: normal subgroup
View other properties obtained by applying the diagonal-in-square operator

A group G is an abelian group if and only if, in the external direct product G \times G, the diagonal subgroup \{ (g,g) \mid g \in G \} is a normal subgroup.

Testing

The testing problem

Further information: Abelianness testing problem

The abelianness testing problem is the problem of testing whether a group (described using some group description rule, such as an encoding of a group or a multi-encoding of a group) is abelian.

Algorithms for the abelianness testing problem range from the brute-force black-box group algorithm for abelianness testing (that involves testing for every pair of elements whether they commute, and is quadratic in the order of the group) to the generating set-based black-box group algorithm for abelianness testing (that involves testing only on a generating set, and is quadratic in the size of the generating set).

GAP command

This group property can be tested using built-in functionality of Groups, Algorithms, Programming (GAP).
The GAP command for this group property is:IsAbelian
The class of all groups with this property can be referred to with the built-in command: AbelianGroups
View GAP-testable group properties

To test whether a group is abelian, the GAP syntax is:

IsAbelian (group)

where group either defines the group or gives the name to a group previously defined.

Study of this notion

Mathematical subject classification

Under the Mathematical subject classification, the study of this notion comes under the class: 20K

References

Textbook references

Book Page number Chapter and section Contextual information View
Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347More info 17 Formal definition (definition as point (2) in general definition of group)
Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261More info 2 1.1 (Rudiments of Group Theory/Review) definition introduced in paragraph Google Books
Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632More info 42 definition introduced in paragraph (immediately after definition of group)
Topics in Algebra by I. N. HersteinMore info 28 Formal definition
A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613More info 2 1.1 (Binary Operations, Semigroups, and Groups) formal definition Google Books
Finite Group Theory (Cambridge Studies in Advanced Mathematics) by Michael Aschbacher, ISBN 0521786754More info 1 1.1 (Elementary group theory) definition introduced in paragraph Google Books

External links

Definition links

Perspective links