A **group** is a mathematical structure consisting of 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.

## Examples

- the set of integers under addition: here, zero is the identity, and the inverse of an element is .
- the set of the positive rational numbers under multiplication: is the identity, while the inverse of an element is .
- the Klein four group consists of the set of formal symbols with the relations All elements of the Klein four group (except the identity 1) have order 2. The Klein four group is isomorphic to under mod addition.
- the set of complex numbers {1, -1,
*i*,*-i*} under multiplication, where*i*is the square root of -1, the basis of the imaginary numbers. This group is isomorphic to under mod addition.

Groups are the appropriate mathematical structures for any application involving symmetry.