# Difference between revisions of "Abelian group"

(→Definition links) |
(→Weaker properties) |
||

(45 intermediate revisions by the same user not shown) | |||

Line 1: | Line 1: | ||

+ | {{basicdef}} | ||

+ | |||

{{pivotal group property}} | {{pivotal group property}} | ||

− | |||

− | |||

==History== | ==History== | ||

Line 7: | Line 7: | ||

===Origin of the term=== | ===Origin of the term=== | ||

− | 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''. | ||

+ | |||

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

+ | <section begin=beginner/> | ||

==Definition== | ==Definition== | ||

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

+ | |||

+ | <center>{{#widget:YouTube|id=uMVm9oSoa6A}}</center> | ||

+ | |||

+ | <section end=beginner/> | ||

− | + | ===Full definition=== | |

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

− | + | * For any <math>a,b,c \in G</math>, <math>a + (b + c) = (a + b) + c</math>. This property is termed [[associativity]]. | |

+ | * For any <math>a \in G</math>, <math>a + 0 = 0 + a = a</math>. <math>0</math> thus plays the role of an additive [[identity element]] or [[neutral element]]. | ||

+ | * For any <math>a \in G</math>, <math>a + (-a) = (-a) + a = 0</math>. Thus, <math>-a</math> is an [[inverse element]] to <math>a</math> with respect to <math>+</math>. | ||

+ | * For any <math>a,b \in G</math>, <math>a + b = b + a</math>. This property is termed [[commutativity]]. | ||

===Equivalent formulations=== | ===Equivalent formulations=== | ||

− | + | A group <math>G</math> is termed abelian if it satisfies the following equivalent conditions: | |

− | * | + | |

+ | * 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/> | ||

+ | |||

+ | ==Notation== | ||

+ | |||

+ | When <math>G</math> 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''. | ||

+ | |||

+ | # The infix operator <math>+</math> is used for the group multiplication, so the sum of two elements <math>a</math> and <math>b</math> is denoted by <math>a + b</math>. The group multiplication is termed ''addition'' and the product of two elements is termed the ''sum''. | ||

+ | # The identity element is typically denoted as <math>0</math> and termed ''zero'' | ||

+ | # The inverse of an element is termed its ''negative'' or ''additive inverse''. The inverse of <math>a</math> is denoted <math>-a</math> | ||

+ | # <math>a + a + \ldots + a</math> done <math>n</math> times is denoted <math>na</math>, (where <math>n \in \mathbb{N}</math>) while <math>(-a) + (-a) + (-a) + \ldots + (-a)</math> done <math>n</math> times is denoted <math>(-n)a</math>. | ||

+ | |||

+ | This convention is typically followed in a situation where we are dealing with the abelian group <math>G</math> 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== | ||

+ | |||

+ | {{group property see examples}} | ||

+ | |||

+ | ===Some infinite examples=== | ||

+ | |||

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

+ | |||

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

+ | <section end=beginner/> | ||

+ | <section begin=revisit/> | ||

+ | ===Finite examples=== | ||

+ | [[Cyclic group]]s are good examples of abelian groups, where the cyclic group of order <math>n</math> is the group of integers modulo <math>n</math>. | ||

+ | |||

+ | Further, any direct product of cyclic groups is also an abelian group. Further, every [[finitely generated group|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 [[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/> | ||

==Facts== | ==Facts== | ||

Line 28: | Line 79: | ||

===Occurrence as subgroups=== | ===Occurrence as subgroups=== | ||

− | Every [[cyclic group]] is | + | 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 subgroup]]s? 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=== | ===Occurrence as quotients=== | ||

− | The maximal | + | 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== | ==Metaproperties== | ||

− | {{ | + | {| class="sortable" border="1" |

+ | ! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols | ||

+ | |- | ||

+ | | [[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== | ||

− | + | ===Stronger properties=== | |

− | {{ | + | {| class="sortable" border="1" |

+ | ! 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 short|abelian group|cyclic 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::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::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=== | |

+ | {| class="sortable" border="1" | ||

+ | ! 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::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::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::FZ-group]] || center has finite index || || {{strictness examples|FZ-group|abelian group}} || {{intermediate notions short|FZ-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}} | ||

+ | |} | ||

− | + | ===Incomparable properties=== | |

− | + | * [[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. | ||

− | + | ==Formalisms== | |

− | A [[direct product]] | + | {{obtainedbyapplyingthe|diagonal-in-square operator|normal subgroup}} |

+ | |||

+ | 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]]. | ||

==Testing== | ==Testing== | ||

Line 58: | Line 155: | ||

{{further|[[Abelianness testing problem]]}} | {{further|[[Abelianness testing problem]]}} | ||

− | {{GAP command for gp|IsAbelian}} | + | 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| | ||

+ | test = IsAbelian| | ||

+ | class = AbelianGroups}} | ||

+ | |||

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

+ | |||

+ | <tt>IsAbelian (group)</tt> | ||

+ | |||

+ | where <tt>group</tt> either defines the group or gives the name to a group previously defined. | ||

+ | |||

+ | ==Study of this notion== | ||

+ | |||

+ | {{msc class|20K}} | ||

− | + | ==References== | |

+ | ===Textbook references=== | ||

− | + | {| class="sortable" border="1" | |

+ | ! Book !! Page number !! Chapter and section !! Contextual information !! View | ||

+ | |- | ||

+ | | {{booklink-defined-tabular|DummitFoote|17|Formal definition (definition as point (2) in general definition of group)|}} || | ||

+ | |- | ||

+ | | {{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== | |

===Definition links=== | ===Definition links=== | ||

− | * {{wp| | + | * {{wp-defined|Abelian group}} |

− | * {{planetmath|AbelianGroup2}} | + | * {{planetmath-defined|AbelianGroup2}} |

* {{mathworld|AbelianGroup}} | * {{mathworld|AbelianGroup}} | ||

− | * {{sor|A/a010230.htm}} | + | * {{sor-defined|A/a010230.htm}} |

===Perspective links=== | ===Perspective links=== | ||

* {{chapman|Abelian_groups}} | * {{chapman|Abelian_groups}} |

## 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: (factscloselyrelated 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 is termed **abelian** if for any elements and in , (here denotes the product of and in ). Note that 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 equipped with a (infix) binary operation (called the addition or group operation), an identity element and a (prefix) unary operation , called the inverse map or negation map, satisfying the following:

- For any , . This property is termed associativity.
- For any , . thus plays the role of an additive identity element or neutral element.
- For any , . Thus, is an inverse element to with respect to .
- For any , . This property is termed commutativity.

### Equivalent formulations

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

- Its center is the whole group.
- Its derived subgroup is trivial.
- (Choose a generating set for ). For any elements , .
- The diagonal subgroup is a normal subgroup inside .

## Notation

When 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*.

- The infix operator is used for the group multiplication, so the sum of two elements and is denoted by . The group multiplication is termed
*addition*and the product of two elements is termed the*sum*. - The identity element is typically denoted as and termed
*zero* - The inverse of an element is termed its
*negative*or*additive inverse*. The inverse of is denoted - done times is denoted , (where ) while done times is denoted .

This convention is typically followed in a situation where we are dealing with the abelian group 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 propertyVIEW: Related group property satisfactions | Related group property dissatisfactions

### Some infinite examples

The additive group of integers , the additive group of rational numbers , the additive group of real numbers , the multiplicative group of nonzero rationals , and the multiplicative group of nonzero real numbers 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 is the group of integers modulo .

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 is an abelian group and is a subgroup of , then is abelian. |

quotient-closed group property | Yes | abelianness is quotient-closed | If is an abelian group and is a normal subgroup of , the quotient group is abelian. |

direct product-closed group property | Yes | abelianness is direct product-closed | Suppose , are abelian groups. Then, the external direct product is also abelian. |

## Relation with other properties

### Stronger properties

### Weaker properties

### Incomparable properties

- Supersolvable group is a group that has a normal series where all the successive quotient groups are cyclic groups. An abelian group is supersolvable if and only if it is finitely generated.
- Polycyclic group is a group that has a subnormal series where all the successive quotent groups are cyclic groups. An abelian group is polycyclic if and only if it is finitely generated.

## 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 is an abelian group if and only if, in the external direct product , the diagonal subgroup 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 ofGroups, 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 978-0471433347^{More 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 0387945261^{More info} |
2 | 1.1 (Rudiments of Group Theory/Review) | definition introduced in paragraph | Google Books |

Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632^{More info} |
42 | definition introduced in paragraph (immediately after definition of group) | ||

Topics in Algebra by I. N. Herstein^{More info} |
28 | Formal definition | ||

A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613^{More 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 0521786754^{More info} |
1 | 1.1 (Elementary group theory) | definition introduced in paragraph | Google Books |