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===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.}}

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===Symbol-free definition===

An '''~~Abelian ~~abelian group''' is a [[group]] where any two elements commute.

===Definition with symbols===

A [[group]] <math>G</math> is termed '''~~Abelian~~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>).

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===Equivalent formulations===

* A group is ~~Abelian ~~abelian if its [[defining ingredient::center]] is the whole group.* A group is ~~Abelian ~~abelian if its [[defining ingredient::commutator subgroup]] is trivial.

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==Notation==

When <math>G</math> is an ~~Abelian ~~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''.

# <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 ~~abelian group <math>G</math> in isolation, rather than as a subgroup of a possibly non-~~Abelian ~~abelian group. If we are working with subgroups in a non-~~Abelian ~~abelian group, we typically use multiplicative notation even if the subgroup happens to be ~~Abelian~~abelian.

==Examples==

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===Finite examples===

[[Cyclic group]]s are good examples of ~~Abelian ~~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 ~~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 ~~abelian groups]].

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

===Non-examples===

Not every group is ~~Abelian~~abelian. The smallest non-~~Abelian ~~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 ~~abelian hinges on the fact that the order in which permutations are performed, matters.

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==Facts==

===Occurrence as subgroups===

Every [[cyclic group]] is ~~Abelian~~abelian. 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 [[~~Abelianization~~abelianization]], and this is the quotient by the [[commutator subgroup]]. A subgroup is normal with ~~Abelian ~~abelian quotient group if and only if the subgroup contains the commutator subgroup.

==Metaproperties==

{{S-closed}}

Any [[subgroup]] of an ~~Abelian ~~abelian group is ~~Abelian ~~abelian -- viz., the property of being ~~Abelian ~~abelian is [[subgroup-closed group property|subgroup-closed]]. This follows as a direct consequence of ~~Abelianness ~~abelianness being varietal. {{proofat|[[Abelianness is subgroup-closed]]}}

{{Q-closed}}

Any [[quotient]] of an ~~Abelian ~~abelian group is ~~Abelian ~~abelian -- viz the property of being ~~Abelian ~~abelian is [[quotient-closed group property|quotient-closed]]. This again follows as a direct consequence of ~~Abelianness ~~abelianness being varietal. {{proofat|[[Abelianness is quotient-closed]]}}

{{DP-closed}}

A [[direct product]] of ~~Abelian ~~abelian groups is ~~Abelian ~~abelian -- viz the property of being ~~Abelian ~~abelian is [[direct product-closed group property|direct product-closed]]. This again follows as a direct consequence of ~~Abelianness ~~abelianness being varietal. {{proofat|[[Abelianness is direct product-closed]]}}

==Testing==

class = AbelianGroups}}

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

<pre>IsAbelian (group)</pre>

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