Abelian Group - Definition

Definition

An abelian group is a set, A, together with an operation "•" that combines any two elements a and b to form another element denoted a • b. The symbol "•" is a general placeholder for a concretely given operation. To qualify as an abelian group, the set and operation, (A, •), must satisfy five requirements 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) holds.

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.

More compactly, 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".