# General semilinear group

## Contents

## Definition

Suppose is a commutative unital ring and is a natural number. The **general semilinear group** of degree over , denoted , is defined as the group of all invertible -semilinear transformations from a -dimensional free module over to itself.

It can also be described explicitly as an external semidirect product of the general linear group by the automorphism group of as a commutative unital ring (acting entry-wise on the matrices), i.e.:

### Special case of fields

Suppose is the prime subfield of and suppose that is a Galois extension over (this is *always* true for a finite field). Then, and we can rewrite the group as:

where the action of on is obtained by inducing the corresponding Galois automorphism on each matrix entry.

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

## Arithmetic functions

### For a finite field

We consider here a field of size where is the field characteristic, so is a natural number.

The prime subfield is , and the extension has degree . The Galois group of the extension thus has size . Note that the Galois group of the extension is always a cyclic group of order and is generated by the Frobenius automorphism .

We are interested in the group .

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

order | order of semidirect product is product of orders: we multiply the order of the Galois group with the order of the general linear group. |

## Particular cases

### Finite cases

We consider a field of size where is the underlying prime and field characteristic, and therefore is the degree of the extension over the prime subfield and also the order of the Galois group.

Note that in the case , the general semilinear group coincides with the general linear group.

(field size) | (underlying prime, field characteristic) | (degree of extension over prime subfield) | order of | ||
---|---|---|---|---|---|

2 | 2 | 1 | 1 | trivial group | 1 |

3 | 3 | 1 | 1 | cyclic group:Z2 | 1 |

4 | 2 | 2 | 1 | symmetric group:S3 | 6 |

5 | 5 | 1 | 1 | cyclic group:Z4 | 4 |

7 | 7 | 1 | 1 | cyclic group:Z6 | 6 |

8 | 2 | 3 | 1 | semidirect product of Z7 and Z3 | 21 |

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

3 | 3 | 1 | 2 | general linear group:GL(2,3) | 48 |

4 | 2 | 2 | 2 | general semilinear group:GammaL(2,4) | 360 |

5 | 5 | 1 | 2 | general linear group:GL(2,5) | 480 |