Difference between revisions of "Abelian group"
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An '''abelian group''' is a [[group]] in which the group [[operation]] is [[commutative]]. | An '''abelian group''' is a [[group]] in which the group [[operation]] is [[commutative]]. | ||
− | For a [[group]] to be considered | + | For a [[group]] to be considered '''abelian''', it must meet several requirements. |
Closure | Closure |
Revision as of 18:52, 12 August 2015
An abelian group is a group in which the group operation is commutative. For a group to be considered abelian, it must meet several requirements.
Closure
For all![]()
![]()
, and for all operations
,
.
Associativity
For all![]()
![]()
and all operations
,
.
Identity Element
There exists somesuch that
.
Inverse Element
For all, there exists some
such that
![]()
Commutativity
For all,
.
A simple example of an abelian group is under addition. It is simple to show that it meets all the requirements.
Closure
For all.
Associativity
For all.
Identity Element
For all.
Inverse Element
For all.
Commutativity
For all.
Seeing as meets all of these requirements under addition, we can say that
is abelian under addition.
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