Tour:Abelian group: Difference between revisions

Revision as of 13:50, 9 May 2008

This article adapts material from the main article: Abelian group

This page is part of the Groupprops Guided tour for beginners (Jump to beginning of tour)
PREVIOUS: Group |UP: Introduction one (beginners) | NEXT: Subgroup

The notion of Abelian group is very important. Abelian groups are those groups where the binary operation is commutative. Read, and thoroughly understand, the definition of Abelian group given below, and then proceed.
Some forms of the definition rely on more advanced terminology and notions; ignore them if they are confusing.
Proceed to Guided tour for beginners:Subgroup, return to Guided tour for beginners:Group or view the full article on Abelian group

@@ Line 4: / Line 4: @@
 {{quotation|The notion of [[Abelian group]] is very important. Abelian groups are those groups where the binary operation is commutative. Read, and thoroughly understand, the definition of Abelian group given below, and then proceed.<br>Some forms of the definition rely on more advanced terminology and notions; ignore them if they are confusing.<br>Proceed to [[Guided tour for beginners:Subgroup]], return to [[Guided tour for beginners:Group]] or [[Abelian group|view the full article on Abelian group]]}}
-==Definition==
+{{#lst:Abelian group|main}}
-===Symbol-free definition===
-An '''Abelian group''' is a [[group]] where any two elements commute.
-===Definition with 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>).
-===Equivalent formulations===
-* A group is Abelian if its [[center]] is the whole group.
-* A group is Abelian if its [[commutator subgroup]] is trivial.
-==Examples==
-[[Cyclic group]]s are good examples of Abelian groups. 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]].