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 | + | For a [[group]] to be considered '''abelian''', it must meet several requirements. |
Closure | Closure | ||
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For all <math>a,b,c</math> <math>\in</math> <math>S</math> and all operations <math>\bullet</math>, <math>(a\bullet b)\bullet c=a\bullet(b\bullet c)</math>. | For all <math>a,b,c</math> <math>\in</math> <math>S</math> and all operations <math>\bullet</math>, <math>(a\bullet b)\bullet c=a\bullet(b\bullet c)</math>. | ||
Identity Element | Identity Element | ||
| − | There exists some <math>e \in S</math> such that <math>a \bullet e = e \bullet a</math>=<math>a</math>. | + | There exists some <math>e \in S</math> such that <math>a \bullet e = e \bullet a = a</math>. |
| + | Inverse Element | ||
| + | For all <math>a \in S</math>, there exists some <math>a^{-1}</math> such that <math>a \bullet a^{-1} = e</math> | ||
| + | Commutativity | ||
| + | For all <math>a,b \in S</math>, <math>a \bullet b = b \bullet a</math>. | ||
| + | A simple example of an abelian group is <math>\mathbb{Z}</math> under addition. It is simple to show that it meets all the requirements. | ||
| + | Closure | ||
| + | For all <math>a,b \in \mathbb{Z} , a+b \in \mathbb{Z}</math>. | ||
| + | Associativity | ||
| + | For all <math>a,b,c \in \mathbb{Z} , (a+b)+c = a+(b+c)</math>. | ||
| + | Identity Element | ||
| + | For all <math>a \in \mathbb{Z} , a+0 = 0+a = a</math>. | ||
| + | Inverse Element | ||
| + | For all <math>a \in \mathbb{Z} , a+ -a = 0</math>. | ||
| + | Commutativity | ||
| + | For all <math>a,b \in \mathbb{Z} , a+b = b+a</math>. | ||
| + | |||
| + | 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 ''not'' abelian because matrix multiplication is associative and not commutative. The smallest finite non-abelian group is the dihedral group of order 6. | ||
| − | |||
[[Category:Definition]] | [[Category:Definition]] | ||
[[Category:Group theory]] | [[Category:Group theory]] | ||
Latest revision as of 17:30, 14 June 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
,
.
, and for all operations 
, 
.
Associativity
For all
.
.
Identity Element
There exists somesuch that
.
such that 
.
Inverse Element
For all, there exists some
such that
, 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 not abelian because matrix multiplication is associative and not commutative. The smallest finite non-abelian group is the dihedral group of order 6.
