Abelian group

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’.

Definition

Template:See also An abelian group is a set, A, together with an operation "•". It combines any two elements a and b to form another element denoted Template:Nowrap. For the group to be abelian, the operation and the elements Template:Nowrap 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 Template:Nowrap 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.

Examples

One example of an abelian group is the set of the integers with the operation of addition. We often write this as $(\mathbb{Z},+)$, where $\mathbb{Z}$ means the set of all integers. This is an abelian group because $(\mathbb{Z},+)$ is a group, and also for any integers $a$ and $b$, the equation $a+b=b+a$ is true. For example, $3+6=6+3$, because both sides are equal to $9$.