# Projective semilinear group of degree two

## Definition

Suppose is a field. The **projective semilinear group of degree two** over is defined as the projective semilinear group of degree two over . It is denoted .

It can be described as an external semidirect product of the projective general linear group of degree two over by the Galois group of over its prime subfield , where the latter acts on the former by applying the Galois automorphism to all the matrix entries in any representing matrix:

In the particular case that is a prime field (i.e., either a field of prime size or the field of rational numbers), can be identified with .

For a prime power , we denote by the group , where is the (unique up to isomorphism) field of size .

## Arithmetic functions

### Over finite field

We consider the case where is the (unique up to isomorphism) field of size , with , so is the field characteristic and is the order of the Galois group .

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

order | -- | order of semidirect product is product of orders: the order of is and the order of is . |

## Particular cases

(field size) | (underlying prime, field characteristic) | (size of Galois group) | Order of (= ) | |
---|---|---|---|---|

2 | 2 | 1 | symmetric group:S3 | 6 |

3 | 3 | 1 | symmetric group:S4 | 24 |

4 | 2 | 2 | symmetric group:S5 | 120 |

5 | 5 | 1 | symmetric group:S5 | 120 |

7 | 7 | 1 | projective general linear group:PGL(2,7) | 336 |

8 | 2 | 3 | Ree group:Ree(3) | 1512 |

9 | 3 | 2 | automorphism group of alternating group:A6 | 1440 |

11 | 11 | 1 | projective general linear group:PGL(2,11) | 1320 |