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Abelian group

10,912 bytes added, 15:35, 11 April 2017
Weaker properties
{{basicdef}}
 
{{pivotal group property}}
 
{{basicdef}}
==History==
===Origin of the term===
The term '''Abelian 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==
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==Symbol-free definition===uMVm9oSoa6A}}</center>
An '''Abelian group''' is a group where any two elements commute.<section end=beginner/>
===Definition with symbolsFull definition===
A 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 '''Abelian''' if for [[associativity]].* For any elements <math>xa \in G</math> and , <math>a + 0 = 0 + a = a</math>. <math>y0</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>xy a + b = yxb + a</math>. This property is termed [[commutativity]].
===Equivalent formulations===
* A group <math>G</math> is Abelian termed abelian if its it satisfies the following equivalent conditions: * Its [[defining ingredient::center]] <math>Z(G)</math> is the whole group.* A 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 if its 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: [[commutator subgroupclassification of finite Abelian groups]] . ===Non-examples=== Not every group is trivialabelian. 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==
===Occurrence as subgroups===
Every [[cyclic group]] is Abelianabelian. Since each group is generated by its cyclic subgroups, every group is generated by a family of Abelian abelian subgroups. A trickier question is: do there exist Abelian abelian [[normal subgroup]]s? A good candidate for an Abelian 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 abelian quotient of any group is termed its [[Abelianizationabelianization]], and this is the quotient by the [[commutator derived subgroup]]. A subgroup is an [[abelian-quotient subgroup]] (i.e., normal with Abelian abelian quotient group ) if and only if the subgroup contains the commutator derived subgroup.
==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_{varietali \in I}G_i</math> is also abelian.|}
Abelian groups form a [[variety of algebras]]. The defining equations for this variety are the equations for a [[group]] along ==Relation with the commutativity equation.other properties==
{{S-closed}}===Stronger properties===
Any {| 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}} |||-| [[subgroupweaker than::homocyclic group]] || direct product of an Abelian isomorphic cyclic groups || || {{strictness examples|abelian group|homocyclic group}} || {{intermediate notions short|abelian group is Abelian |homocyclic group}}|||-| [[Weaker than::residually cyclic group]] || every non- viz the property of being Abelian 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 subgroupis 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}} || |-closed | [[weaker than::finite abelian group]] || abelian and a [[finite group]] || || {{strictness examples|abelian group|finite group}} || {{intermediate notions short|abelian group property|subgroupfinite abelian group}}|||-closed| [[weaker than::finitely generated abelian group]]. This follows as || abelian and a direct consequence of Abelianness being varietal.[[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}} || {{Qintermediate notions short|nilpotent group|abelian group}}|-closed| [[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}}|}
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.===Incomparable properties===
{{DP-closed}}* [[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== {{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]] of Abelian groups is Abelian -- viz <math>G \times G</math>, the property of being Abelian diagonal subgroup <math>\{ (g,g) \mid g \in G \}</math> is a [[direct product-closed group property|direct product-closednormal subgroup]]. This again follows as a direct consequence of Abelianness being varietal.
==Testing==
{{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|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==
To test whether a group is Abelian, the GAP syntax is:{{msc class|20K}}
<pre>IsAbelian (group)</pre>==References=====Textbook references===
where <pre>{| 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/pre> either defines the books?id=EroGCAAAQBAJ&pg=PA2 Google Books]|-| {{booklink-defined-tabular|Artin|42||definition introduced in paragraph (immediately after definition of group or gives the name to a )}} |||-| {{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 previously definedtheory)|definition introduced in paragraph}} || [https://books.google.com/books?id=BprbtnlI6HEC&pg=PA1 Google Books]|}
==External links==
===Definition links===
* {{wp-defined|Abelian_groupAbelian group}}* {{planetmath-defined|AbelianGroup2}}
* {{mathworld|AbelianGroup}}
* {{sor-defined|A/a010230.htm}}
===Perspective links===
* {{chapman|Abelian_groups}}
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