# Changes

## Abelian group

, 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]] $G$ 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. <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 $G$ equipped with a (infix) binary operation $+$ (called the addition or group operation), an identity element $0$ and a (prefix) unary operation $-</math>, called the inverse map or negation map, satisfying the following: * For any [itex]a,b,c \in G$ , $a + (b + c) = (a + b) + c$. This property is termed '''Abelian''' if for [[associativity]].* For any elements $xa \in G$ and , $a + 0 = 0 + a = a$. $y0$ thus plays the role of an additive [[identity element]] or [[neutral element]].* For any $a \in G$, $a + (-a) = (-a) + a = 0$. Thus, $-a$ is an [[inverse element]] to $a$ with respect to $+</math>.* For any [itex]a,b \in G$, $xy a + b = yxb + a$. This property is termed [[commutativity]].
===Equivalent formulations===
* A group $G$ is Abelian termed abelian if its it satisfies the following equivalent conditions: * Its [[defining ingredient::center]] $Z(G)$ is the whole group.* A Its [[defining ingredient::derived subgroup]] $G' = [G,G]$ is trivial.* (Choose a generating set $S$ for $G$). For any elements $a,b \in S$, $ab = ba$.* The diagonal subgroup $\{ (g,g) \mid g \in G \}$ is a [[defining ingredient::normal subgroup]] inside $G \times G$.<section begin=beginner/> ==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''.  # 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''. # The identity element is typically denoted as $0$ and termed ''zero''# The inverse of an element is termed its ''negative'' or ''additive inverse''. The inverse of $a$ is denoted $-a$# $a + a + \ldots + a$ 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== {{group property see examples}} ===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).<section end=beginner/><section begin=revisit/>===Finite examples===[[Cyclic group]]s 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 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 $G$ is an abelian group and $H$ is a subgroup of $G$, then $H$ is abelian.|-| [[satisfies metaproperty::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.|-| [[satisfies metaproperty::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 $\prod_{varietali \in I}G_i$ 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 $G$ is an abelian group if and only if, in the [[external direct product]] of Abelian groups is Abelian -- viz $G \times G$, the property of being Abelian diagonal subgroup $\{ (g,g) \mid g \in G \}$ 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===