# Field:F3

From Groupprops

## Definition

This field, denoted or , is the unique field of three elements. It can be defined as the ring of integers modulo .

## Related groups

Group functor | Value | GAP ID |
---|---|---|

additive group | cyclic group:Z3 | (3,1) |

multiplicative group | cyclic group:Z2 | (2,1) |

general affine group of degree one | symmetric group:S3 | (6,1) |

general linear group of degree two | general linear group:GL(2,3) | (48,29) |

special linear group of degree two | special linear group:SL(2,3) | (24,3) |

projective general linear group of degree two | symmetric group:S4 | (24,12) |

projective special linear group of degree two | alternating group:A4 | (12,3) |

projective special linear group of degree three | projective special linear group:PSL(3,3) | order 5616, no GAP ID. |

upper-triangular unipotent matrix group of degree three | prime-cube order group:U(3,3) | (27,3) |

## GAP implementation

The field can be defined using GAP's GF function as:

`GF(3)`

It can also be defined using GAP's ZmodnZ function as:

`ZmodnZ(3)`