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]]. They are named after Norwegian mathematician Niels Abel. |
For a [[group]] to be considered '''abelian''', it must meet several requirements. | For a [[group]] to be considered '''abelian''', it must meet several requirements. | ||
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Seeing as <math>\mathbb{Z}</math> meets all of these requirements under addition, we can say that <math>\mathbb{Z}</math> is abelian under addition. | Seeing as <math>\mathbb{Z}</math> meets all of these requirements under addition, we can say that <math>\mathbb{Z}</math> is abelian under addition. | ||
+ | ==Examples== | ||
+ | Notable examples of Abelian groups include the integers under addition, the real numbers under addition, the integers modulo <math>n</math> under addition, the multiplicative group of integers modulo <math>n</math>, and the additive group of any ring. Many matrix groups are \textit{not} Abelian because matrix multiplication is associative and not commutative. The smallest finite non-Abelian group is the dihedral group of order 6. | ||
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Revision as of 18:17, 25 January 2020
An abelian group is a group in which the group operation is commutative. They are named after Norwegian mathematician Niels Abel. 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.
Examples
Notable examples of Abelian groups include the integers under addition, the real numbers under addition, the integers modulo under addition, the multiplicative group of integers modulo
, and the additive group of any ring. Many matrix groups are \textit{not} Abelian because matrix multiplication is associative and not commutative. The smallest finite non-Abelian group is the dihedral group of order 6.
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