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, the operation must be commutative, meaning that the order in which two elements are combined does not affect the result. In an Abelian group, for any two elements a and b, the equation a + b = b + a holds true. Common examples of Abelian groups include the set of integers under addition and the set of real numbers under addition. These structures are fundamental in various areas of abstract algebra and have applications in fields such as cryptography and physics.