groepsoperatie

A groepsoperatie is the Dutch term for a group operation in the mathematical field of abstract algebra. It refers to a binary operation defined on a set that equips the set with the structure of a group. The definition requires four properties: closure, associativity, the existence of an identity element, and the existence of inverses for all elements. In more formal terms, a group (G,·) consists of a non‑empty set G together with a binary operation ·: G × G → G such that for all a, b, c in G the following hold. Closure: a·b is in G. Associativity: a·(b·c) = (a·b)·c. Identity: there exists an element e in G satisfying e·a = a·e = a for all a in G. Inverses: for each a in G there exists an element a⁻¹ in G with a·a⁻¹ = a⁻¹·a = e.

Good examples of groepsoperaties include addition on the set of integers, multiplication on non‑zero rational numbers, and permutation composition in symmetric groups. Group operations can be commutative, in which case the group is called abelian; otherwise it is non‑abelian. The study of groepsoperaties underpins much of modern algebra, including the classification of finite groups, representation theory, and applications to geometry and number theory. The term is widely used in Dutch mathematical literature, and translations often include "group operation" or "binary operation" depending on context. All definitions and properties remain identical across languages; only terminology differs.