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

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

## 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.

$q$ (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 trivial group 1
4 2 2 1 cyclic group:Z2 1
5 5 1 1 trivial group 1
7 7 1 1 trivial group 1
8 2 3 1 cyclic group:Z3 3
9 3 2 1 cyclic group:Z2 1
2 2 1 2 symmetric group:S3 6
3 3 1 2 symmetric group:S4 24
4 2 2 2 symmetric group:S5 120
5 5 1 2 symmetric group:S5 120
7 7 1 2 projective general linear group:PGL(2,7) 336
8 2 3 2 Ree group:Ree(3) 1512
9 3 2 2 automorphism group of alternating group:A6 1440