grouplaw

In mathematics, the group law (often called the group operation) is the binary operation that endows a set with a group structure. Formally, a group is a set G equipped with a binary operation *: G × G → G that assigns to every pair (a, b) an element a*b of G and satisfies four axioms: closure, associativity, identity, and inverses.

Closure means that applying the operation to any two elements of G yields another element of G. Associativity requires that (a*b)*c = a*(b*c) for all a, b, c in G. The identity axiom asserts there exists an element e in G such that e*a = a*e = a for all a in G. The inverse axiom states that every element a in G has an inverse b in G with a*b = b*a = e.

The operation is usually denoted multiplicatively as ab, additively as a+b, or with a symbol such as ⋆. The choice of notation may reflect the mathematical structure (for example, additive groups or multiplicative groups).

Examples include the integers under addition (Z, +), whose identity is 0 and inverses are negatives; the nonzero real numbers under multiplication (R\{0}, ×), with identity 1 and inverses 1/x; and the symmetric group S_n under composition of functions, with the identity being the identity permutation.

Variants include abelian (commutative) groups, non-abelian groups, Lie groups with smooth operations, and topological groups where the binary operation is continuous. The group law is central to abstract algebra and underlies many structures in mathematics, physics, and computer science.