# Abelian group

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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 $a,b$ $\in$ $S$, and for all operations $\bullet$, $a\bullet b \in S$.


Associativity

         For all $a,b,c$ $\in$ $S$ and all operations $\bullet$, $(a\bullet b)\bullet c=a\bullet(b\bullet c)$.


Identity Element

         There exists some $e \in S$ such that $a \bullet e = e \bullet a = a$.


Inverse Element

         For all $a \in S$, there exists some $a^{-1}$ such that $a \bullet a^{-1} = e$


Commutativity

         For all $a,b \in S$, $a \bullet b = b \bullet a$.


A simple example of an abelian group is $\mathbb{Z}$ under addition. It is simple to show that it meets all the requirements.

Closure

         For all $a,b \in \mathbb{Z} , a+b \in \mathbb{Z}$.


Associativity

         For all $a,b,c \in \mathbb{Z} , (a+b)+c = a+(b+c)$.


Identity Element

         For all $a \in \mathbb{Z} , a+0 = 0+a = a$.


Inverse Element

         For all $a \in \mathbb{Z} , a+ -a = 0$.


Commutativity

         For all $a,b \in \mathbb{Z} , a+b = b+a$.


Seeing as $\mathbb{Z}$ meets all of these requirements under addition, we can say that $\mathbb{Z}$ is abelian under addition.

## Examples

Notable examples of abelian groups include the integers under addition, the real numbers under addition, the integers modulo $n$ under addition, the multiplicative group of integers modulo $n$, 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.