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Abelian group

, 23:23, 21 January 2009
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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]] $G$ is termed '''Abelianabelian''' if for any elements $x$ and $y$ in $G$, $xy = yx$ (here $xy$ denotes the product of $x$ and $y$ in $G$).
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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 $G$ 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 $+$ 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''.
# $a + a + \ldots + a$ 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 abelian group $G$ 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 Abelianabelian.
==Examples==
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===Finite examples===
[[Cyclic group]]s are good examples of Abelian 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 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 Abelianabelian. 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 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 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 Abelianabelian, the GAP syntax is:
<pre>IsAbelian (group)</pre>