# Symmetric group on finite set

## Contents

## Definition

A **symmetric group on finite set** or **symmetric group of finite degree** is a symmetric group on a finite set.

See symmetric group for more general information about symmetric groups.

## Particular cases

### Small finite values

Since alternating groups are simple for degree at least five, all symmetric groups of degree at least five are *not* solvable. Also, all symmetric groups of degree greater than two are centerless, and among them, the one of degree six is the only one that is not complete.

Cardinality of set | Common name for symmetric group of that degree | Order with prime factorization | Comments |
---|---|---|---|

0 | Trivial group | 1 | Trivial |

1 | Trivial group | 1 | Trivial |

2 | Cyclic group:Z2 | 2 | group of prime order. In particular, abelian |

3 | Symmetric group:S3 | supersolvable but not nilpotent. Also, complete | |

4 | Symmetric group:S4 | solvable but not supersolvable or nilpotent. Also, complete | |

5 | Symmetric group:S5 | not solvable. Has simple non-abelian subgroup of index two. Also, complete | |

6 | Symmetric group:S6 | not solvable, and not complete. | |

7 | Symmetric group:S7 | complete, not solvable, has simple non-abelian subgroup of index two. | |

8 | Symmetric group:S8 | complete, not solvable, has simple non-abelian subgroup of index two. |

## Group properties

Below we discuss properties satisfied for the symmetric group of degree .

Property | Satisfied? | Explanation | Comment |
---|---|---|---|

abelian group | No for | ||

nilpotent group | No for | ||

solvable group | No for | ||

complete group | Yes for | symmetric groups are complete | |

centerless group | Yes for | symmetric groups are centerless | |

ambivalent group | Yes | symmetric groups are ambivalent | |

strongly rational group | Yes | symmetric groups are strongly rational |