# Group (mathematics)

 $\frac{d}{dx} \sin x=?\,$ This article/section deals with mathematical concepts appropriate for a student in late high school or early university.

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: (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 − 1A = I

A group with commutative binary operator is known as Abelian.

## Examples

1. the set of integers $\mathbb{Z}$ under addition, $(\mathbb{Z},+)$: here, zero is the identity, and the inverse of an element $a \in \mathbb{Z}$ is a.
2. the set of the positive rational numbers $\mathbb{Q}_+$ under multiplication, $(\mathbb{Q}_+,\cdot)$: 1 is the identity, while the inverse of an element $\frac{m}{n} \in \mathbb{Q}_+$ is $\frac{n}{m}$.
3. for every $n \in \mathbb{N}$ there exists at least one group with n elements,e.g., $(\mathbb{Z}/n\mathbb{Z},+) = (\mathbb{Z}_n,+).$
4. the set of complex numbers {1, -1, i,-i} under multiplication, where i is the principal square root of -1, the basis of the imaginary numbers. This group is isomorphic to $\mathbb{Z}_{4}$ under mod addition.
5. the Klein four group consists of the set of formal symbols {1,i,j,k} with the relations $i^{2} =j^{2}=k^{2}=1, \; ij=k, \; jk=i, \; ki=j.$ All elements of the Klein four group (except the identity 1) have order 2. The Klein four group is isomorphic to $\mathbb{Z}_{2} \times \mathbb{Z}_{2}$ under mod addition.
6. the set of "moves" on a Rubik's cube, where a move is understood to be a finite sequence of twists: here, the identity move is to do nothing, while the inverse of a move is to do the move in reverse, thereby undoing it.
7. The Symmetric group
8. The general and special Linear groups.

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