Projective semilinear group

Definition
Let $$K$$ be a field and $$n$$ be a natural number.

The projective semilinear group or projective general semilinear group $$P\Gamma L(n,K)$$ is defined as the quotient group of the defining ingredient::general semilinear group $$\Gamma L(n,K)$$ by the subgroup corresponding to scalar linear transformations, i.e., the subgroup corresponding to the center of $$GL(n,K)$$. (Note that the subgroup by which we are quotienting is not in the center of $$\Gamma L(n,K)$$ in general).

It can also be defined as an external semidirect product of the defining ingredient::projective general linear group by a Galois group:

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

where $$k$$ is the prime subfield of $$K$$.

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

Arithmetic functions
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 $$P\Gamma L(n,q)$$.

Particular 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 projective general semilinear group coincides with the projective general linear group.

In the case $$n = 1$$, the projective semilinear group is just the Galois group, which is cyclic of order $$r$$.

Finally, if $$r = n = 1$$, we get a trivial group.