View source for Abelian group

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 <math>a,b</math> <math>\in</math> <math>S</math>, and for all operations <math>\bullet</math>, <math>a\bullet b \in S</math>.
Associativity 
          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
          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:Group theory]]