# Group

A set `G`, together with a biary operation •, which satisfy:

**Closure**:`x`•`y`is in`G`, for all`x`,`y`in`G`.**Identity**: There exists`e`in`G`, such that`x`•`e`=`e`•`x`=`x`, for all`x`in`G`.**Inverses**: For all`x`in`G`, there exists an`x`' in`G`, such that`x`•`x`' =`x`' •`x`=`e`.**Associativity**:`x`• (`y`•`z`) = (`x`•`y`) •`z`, for all`x`,`y`,`z`in`G`.

Technically, `G` is the set and the group itself should be denoted ⟨G,•⟩.
However, for brevity or where the intended operation is obvious, it is common to refer
simply to the group `G`.

A group is said to be **abelian** if all its elements commute; in other words,
if `a` • `b` = `b` • `a` for all `a`,`b` in `G`.

The **order** of a group, `G`, is the number of elements in the set `G`.
It is usually witten as |`G`|, using standard set notation.
When the order of the group is small, there are only a few possibilities for the
structure which the group can have. Groups which have exactly the same structure
are said to be isomorphic to one another.

Order | Abelian Groups | Non-Abelian Groups |
---|---|---|

0 | - | - |

1 | E |
- |

2 | C_{2} |
- |

3 | C_{3} |
- |

4 | C_{4}
; C_{2}×C_{2} |
- |

5 | C_{5} |
- |

6 | C_{6} |
D_{6} |

7 | C_{7} |
- |

8 | C_{8}
; C_{4}×C_{2}
; C_{2}×C_{2}×C_{2} |
D_{8}
; Q |

9 | C_{9}
; C_{3}×C_{3} |
- |

10 | C_{10} |
D_{10} |

11 | C_{11} |
- |

See also:
Alternating Group, `A`_{n};
Cyclic Group, `C`_{n};
Dihedral Group, `D`_{2n};
Symmetric Group, `S`_{n}.