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   , and for all operations , .

Associativity

         For all    and all operations , .

Identity Element

         There exists some  such 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 not abelian because matrix multiplication is associative and not commutative. The smallest finite non-abelian group is the dihedral group of order 6.