# Projective special linear group:PSL(3,3)

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## Contents

## Definition

This group is a finite group defined in the following equivalent ways:

- It is the projective special linear group of degree three over field:F3, denoted .
- It is the special linear group of degree three over field:F3, denoted .
- It is the projective general linear group of degree three over field:F3, denoted .

### Equivalence of definitions

The equivalence of definitions follows from isomorphism between linear groups when degree power map is bijective.

## Arithmetic functions

### Basic arithmetic functions

Function | Value | Similar groups | Explanation |
---|---|---|---|

order (number of elements, equivalently, cardinality or size of underlying set) | 5616 | groups with same order | As : |

exponent of a group | 312 | groups with same order and exponent of a group | groups with same exponent of a group | |

Frattini length | 1 | groups with same order and Frattini length | groups with same Frattini length | As the group is a simple non-abelian group, its Frattini length must be one. |

### Arithmetic functions of a counting nature

Function | Value | Explanation |
---|---|---|

number of subgroups | 6374 | |

number of conjugacy classes | 12 | As : |

number of conjugacy classes of subgroups | 51 |

## Group properties

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

abelian group | No | |

nilpotent group | No | |

solvable group | No | |

simple group, simple non-abelian group | Yes | projective special linear group is simple |

minimal simple group | Yes |

## GAP implementation

Description | Functions used |
---|---|

PSL(3,3) |
PSL |

SL(3,3) |
SL |

PGL(3,3) |
PGL |