# Field:F2

From Groupprops

## Definition

This field, denoted or , is the unique field (up to isomorphism) of two elements.

## Facts

- For any natural number , the projective general linear group, projective special linear group, special linear group, and general linear group are all the same.

## Related groups

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

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

multiplicative group | trivial group | (1,1) |

general affine group of degree one | cyclic group:Z2 | (2,1) |

general linear group of degree two | symmetric group:S3 | (6,1) |

special linear group of degree two | symmetric group:S3 | (6,1) |

projective general linear group of degree two | symmetric group:S3 | (6,1) |

projective special linear group of degree two | symmetric group:S3 | (6,1) |

outer linear group of degree two | direct product of S3 and Z2 | (12,4) |

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

projective special linear group of degree three | projective special linear group:PSL(3,2) | (168,42) |

projective special linear group of degree four | alternating group:A8 | The order is 20160, no GAP ID. |

## GAP implementation

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

`GF(2)`

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

`ZmodnZ(2)`