# Group

A mathematical group is a set of elements and a binary operation that together satisfy the four fundamental properties of closure, associativity, the identity property, and the inverse property.

Group is a college-level concept that would be first encountered in an abstract algebra course covering group theory.

### Examples

Abelian Group: | An Abelian group is a group for which the binary operation is commutative. |

Cyclic Group: | A cyclic graph is an (always Abelian) abstract group generated by a single element. |

Dihedral Group: | The dihedral group of order n>i is the symmetry group for a regular polygon with n sides. |

Finite Group: | A finite group is a group with a finite number of elements. |

Simple Group: | A simple group is a mathematical group whose only normal subgroups are the trivial subgroup of order one and the improper subgroup consisting of the entire original group. |

Symmetric Group: | A symmetric group is a group of all permutations of a given set. |

### Prerequisites

Congruence: | A congruence is an equation in modular arithmetic, i.e., one in which only the remainders relative to some base, known as the "modulus," are significant. |

Matrix: | A matrix is a concise and useful way of uniquely representing and working with linear transformations. In particular, for every linear transformation, there exists exactly one corresponding matrix, and every matrix corresponds to a unique linear transformation. The matrix is an extremely important concept in linear algebra. |

Permutation: | In combinatorics, a permutation is a rearrangement of the elements in an ordered list S into a one-to-one correspondence with S itself. Combinatorics studies the number of possible ways of doing this under various conditions. |