An abelian group is a mathematical structure in which a set of elements is combined with an 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 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. These groups are named after the mathematician Niels Henrik Abel, who contributed significantly to the field of group theory.