# Group (mathematics)

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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:

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

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 −
*a*. - the set of the positive rational numbers under multiplication, : 1 is the identity, while the inverse of an element is .
- for every there exists at least one group with n elements,e.g.,
- 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 under mod addition. - the Klein four group consists of the set of formal symbols {1,
*i*,*j*,*k*} 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 "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.
- The Symmetric group
- The general and special Linear groups.

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