# Projective special linear group

## Particular cases

### Finite fields

Some facts:

- For , . For a power of two, but this is not equal to .
- Projective special linear group equals alternating group in only finitely many cases: All those cases are listed in the table below.
- Projective special linear group is simple except for finitely many cases, all of which are listed below.

Size of field | Order of matrices | Common name for the projective special linear group | Order of group | Comment |
---|---|---|---|---|

1 | Trivial group | Trivial | ||

2 | 2 | Symmetric group:S3 | supersolvable but not nilpotent. Not simple. | |

3 | 2 | Alternating group:A4 | solvable but not supersolvable group. Not simple. | |

4 | 2 | Alternating group:A5 | simple non-abelian group of smallest order. | |

5 | 2 | Alternating group:A5 | simple non-abelian group of smallest order. | |

7 | 2 | Projective special linear group:PSL(3,2) | simple non-abelian group of second smallest order. | |

9 | 2 | Alternating group:A6 | simple non-abelian group. | |

2 | 3 | Projective special linear group:PSL(3,2) | simple non-abelian group of second smallest order. | |

3 | 3 | Projective special linear group:PSL(3,3) | simple non-abelian group. |