Group (mathematics)

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

1. Closure: applying the binary operator to any two elements of the group produces a result which itself belongs to the group
2. Associativity: $(AB)C=A(BC)$ where $A$, $B$ and $C$ are any element of the group
3. Existence of Identity: there must exist an identity element $I$ such that $IA=AI=A$; that is, applying the binary operator to some element $A$ and the identity element $I$ leaves $A$ unchanged
4. Existence of Inverse: for each element $A$, there must exist an inverse $A^{-1}$ such that $AA^{-1}=A^{-1}A=I$

A group with commutative binary operator is known as Abelian.

Example: the Klein Four Group consists of the set of numbers {1, -1, i,-i} under the binary operation of multiplication, where i is the square root of -1, the basis of the imaginary numbers.

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