A **group** is a set of elements combined with a binary operator which satisfies four conditions:

**Closure**: applying the binary operator to any two elements of the group produces a result which itself belongs to the group**Associativity**: where , and are any element of the group**Existence of Identity**: there must exist an identity element such that ; that is, applying the binary operator to some element and the identity element leaves unchanged**Existence of Inverse**: for each element , there must exist an inverse such that

A group with commutative binary operator is known as Abelian.