Group (mathematics)

A group is a mathematical structure consisting of a 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: <math>(AB)C = A(BC)</math> where <math>A</math>, <math>B</math> and <math>C</math> are any element of the group
3. Existence of Identity: there must exist an identity element <math>I</math> such that <math>IA = AI = A</math>; that is, applying the binary operator to some element <math>A</math> and the identity element <math>I</math> leaves <math>A</math> unchanged
4. Existence of Inverse: for each element <math>A</math>, there must exist an inverse <math>A^{-1}</math> such that <math>AA^{-1} = A^{-1}A = I</math>

A group with commutative binary operator is known as Abelian.

Examples

1. the set of integers <math>\mathbb{Z}</math> under addition: here, zero is the identity, and the inverse of an element <math>a \in \mathbb{Z}</math> is <math>-a</math>.
2. the set of the positive rational numbers <math>\mathbb{Q}_+</math> under multiplication: <math>1</math> is the identity, while the inverse of an element <math>\frac{m}{n} \in \mathbb{Q}_+</math> is <math>\frac{n}{m}</math>.
3. the Klein four group consists of the set of formal symbols <math>\{1, i, j, k \} </math> with the relations <math> i^{2} =j^{2}=k^{2}=1, \; ij=k, \; jk=i, \; ki=j. </math> All elements of the Klein four group (except the identity 1) have order 2. The Klein four group is isomorphic to <math>\mathbb{Z}_{2} \times \mathbb{Z}_{2}</math> under mod addition.
4. 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 <math> \mathbb{Z}_{4} </math> under mod addition.

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