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.