Abelian group

In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel. The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified. An abelian group is a set , together with an operation that combines any two elements and of to form another element of denoted . The symbol is a general placeholder for a concretely given operation. To qualify as an abelian group, the set and operation, , must satisfy four requirements known as the abelian group axioms (some authors include in the axioms some properties that belong to the definition of an operation: namely that the operation is defined for any ordered pair of elements of A, that the result is well-defined, and that the result belongs to A): Associativity For all , , and in , the equation holds. Identity element There exists an element in , such that for all elements in , the equation holds. Inverse element For each in there exists an element in such that , where is the identity element. Commutativity For all , in , . A group in which the group operation is not commutative is called a "non-abelian group" or "non-commutative group". Additive group and Multiplicative group There are two main notational conventions for abelian groups – additive and multiplicative. Generally, the multiplicative notation is the usual notation for groups, while the additive notation is the usual notation for modules and rings.