An Abelian group is a mathematical structure consisting of a set equipped with an operation that combines any two elements to form a third element. This operation must satisfy four key properties: closure, associativity, the existence of an identity element, and the existence of inverses. Additionally, an Abelian group requires that the operation is commutative, meaning the order in which two elements are combined does not affect the result. Common examples of Abelian groups include the set of integers under addition and the set of real numbers under addition. The term "Abelian" honors the mathematician Niels Henrik Abel, who contributed significantly to group theory. Abelian groups are fundamental in various areas of mathematics, including abstract algebra and number theory.