Group

An algebraic structure consisting of a set equipped with a single binary operation that is associative, has an identity element, and such that every element has an inverse with respect to the operation.

A set equipped with a single binary operation that is associative, has an identity element, and in which every element has an inverse under that operation.

An algebraic structure (G, ·) consisting of a set G together with a binary operation · that is closed, associative, has an identity element e in G, and for every element g in G an inverse g^{-1} in G. The structure is considered up to equality of the set and operation, and homomorphisms preserve the operation.