Abelian group

An Abelian group is a set of elements with a binary operation that satisfies specific axioms, including closure, associativity, an identity element, and an inverse for each element. Crucially, it also adheres to the commutative law, meaning the order of operations does not affect the outcome. A common example is the set of integers under addition, where the identity element is 0 and each integer has a corresponding negative integer as its inverse.1

Group theory, which studies these structures, is fundamental to modern algebra and has applications in various fields, including geometry, physics, chemistry, and computer science.1