What is an Abelian Group in Mathematics?

In mathematics, particularly in group theory, a group is called Abelian if its operation is commutative, meaning that the order in which elements are combined does not affect the result. In other words, if we have a group G and two of its elements a and b, then the product ab = ba.

For example, the set of integers under addition is an Abelian group, because the order in which we add numbers does not matter: 2 + 3 = 3 + 2. Similarly, the set of real numbers under multiplication is also Abelian, because multiplying numbers in any order gives the same result: (2 x 3) x 4 = 2 x (3 x 4).

The term "Abelian" comes from the name of the mathematician Niels Henrik Abel, who worked on group theory in the early 19th century. It is often used to describe groups that have this particular property of commutativity.