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