An abelian group is a set equipped with a binary operation that satisfies four key properties: closure, associativity, identity, and invertibility. Additionally, in an abelian group, the operation is commutative, meaning that the order in which 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. These groups are named after the mathematician Niels Henrik Abel, who contributed significantly to the field of group theory. Abelian groups are fundamental in various areas of mathematics, including algebra and topology.