# General semilinear group

## Definition

Suppose $R$ is a commutative unital ring and $n$ is a natural number. The general semilinear group of degree $n$ over $R$, denoted $\Gamma L(n,R)$, is defined as the group of all invertible $R$-semilinear transformations from a $n$-dimensional free module over $R$ to itself.

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

$\Gamma L(n,R) = GL(n,R) \rtimes \operatorname{Aut}(R)$

### Special case of fields

Suppose $k$ is the prime subfield of $K$ and suppose that $K$ is a Galois extension over $k$ (this is always true for $K$ a finite field). Then, $\operatorname{Aut}(K) = \operatorname{Gal}(K/k)$ and we can rewrite the group as:

$\Gamma L(n,K) = GL(n,K) \rtimes \operatorname{Gal}(K/k)$

where the action of $\operatorname{Gal}(K/k)$ on $GL(n,K)$ is obtained by inducing the corresponding Galois automorphism on each matrix entry.

If $q$ is a prime power, we denote by $\Gamma L(n,q)$ the group $\Gamma L(n,\mathbb{F}_q)$ where $\mathbb{F}_q$ is the (unique up to isomorphism) field of size $q$.

## Arithmetic functions

### For a finite field

We consider here a field $K = \mathbb{F}_q$ of size $q = p^r$ where $p$ is the field characteristic, so $r$ is a natural number.

The prime subfield is $k = \mathbb{F}_p$, and the extension $K/k$ has degree $r$. The Galois group of the extension thus has size $r$. Note that the Galois group of the extension is always a cyclic group of order $r$ and is generated by the Frobenius automorphism $x \mapsto x^p$.

We are interested in the group $\Gamma L(n,q)$.

Function Value Explanation
order $rq^{\binom{n}{2}}\prod_{i=1}^n (q^i - 1)$ order of semidirect product is product of orders: we multiply the order $r$ of the Galois group with the order of the general linear group.

## Particular cases

### Finite cases

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

Note that in the case $r = 1$, the general semilinear group coincides with the general linear group.

$q$ (field size) $p$ (underlying prime, field characteristic) $r$ (degree of extension over prime subfield) $n$ $\Gamma L(n,q)$ order of $\Gamma L(n,q)$
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