Abelian group

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In group theory, an abelian group is a group that is commutative.

Definition[change | change source]

An abelian group is a set, A, together with an operation "•". It combines any two elements a and b to form another element denoted ab. 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 ab is also in A.
Associativity
For all a, b and c in A, the equation (ab) • c = a • (bc) is true.
Identity element
There exists an element e in A, such that for all elements a in A, the equation ea = ae = a holds.
Inverse element
For each a in A, there exists an element b in A such that ab = ba = e, where e is the identity element.
Commutativity
For all a, b in A, ab = ba.

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