General semilinear group

Definition
Suppose $$K$$ is a field and $$n$$ is a natural number. The general semilinear group of degree $$n$$ over $$K$$, denoted $$\Gamma L(n,k)$$, is defined as the group of all semilinear transformations from a $$n$$-dimensional vector space over $$K$$ to itself.

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

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

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 $$q$$.

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 $$\Gamma L(n,q)$$.

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.