group whose group operation is commutative From Wikipedia, the free encyclopedia

In group theory, an **abelian group** is a group with operation that is commutative. Because of that, an abelian group is sometimes called a ‘commutative group’.

A group in which the group operation is not commutative is called a ‘non-abelian group’ or ‘non-commutative group’.

An abelian group is a set, *A*, together with an operation "•". It combines any two elements *a* and *b* to form another element denoted *a* • *b*. For the group to be abelian, the operation and the elements (*A*, •) must follow some requirements. These are known as the *abelian group axioms*:

- Closure
- For all
*a*,*b*in*A*, the result of the operation*a*•*b*is also in*A*. - Associativity
- For all
*a*,*b*and*c*in*A*, the equation (*a*•*b*) •*c*=*a*• (*b*•*c*) is true. - Identity element
- There exists an element
*e*in*A*, such that for all elements*a*in*A*, the equation*e*•*a*=*a*•*e*=*a*holds. - Inverse element
- For each
*a*in*A*, there exists an element*b*in*A*such that*a*•*b*=*b*•*a*=*e*, where*e*is the identity element. - Commutativity
- For all
*a*,*b*in*A*,*a*•*b*=*b*•*a*.

One example of an abelian group is the set of the integers with the operation of addition. We often write this as , where means the set of all integers. This is an abelian group because is a group, and also for any integers and , the equation is true. For example, , because both sides are equal to .

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