# Projective semilinear group of degree two

## Definition

Suppose $K$ is a field. The projective semilinear group of degree two over $K$ is defined as the projective semilinear group of degree two over $K$. It is denoted $P\Gamma L(2,K)$.

It can be described as an external semidirect product of the projective general linear group of degree two over $K$ by the Galois group of $K$ over its prime subfield $k$, where the latter acts on the former by applying the Galois automorphism to all the matrix entries in any representing matrix: $P\Gamma L(2,K) = PGL(2,K) \rtimes \operatorname{Gal}(K/k)$

In the particular case that $K$ is a prime field (i.e., either a field of prime size or the field of rational numbers), $P\Gamma L(2,K)$ can be identified with $PGL(2,K)$.

For a prime power $q$, we denote by $P\Gamma L(2,q)$ the group $P\Gamma L(2,\mathbb{F}_q)$, where $\mathbb{F}_q$ is the (unique up to isomorphism) field of size $q$.

## Arithmetic functions

### Over finite field

We consider the case where $K$ is the (unique up to isomorphism) field of size $q$, with $q = p^r$, so $p$ is the field characteristic and $r$ is the order of the Galois group $\operatorname{Gal}(K/k)$.

Function Value Similar groups Explanation
order $r(q^3 - q) = rq(q+1)(q-1)$ -- order of semidirect product is product of orders: the order of $PGL(2,q)$ is $q^3 - q$ and the order of $\operatorname{Gal}(K/k)$ is $r$.

## Particular cases $q$ (field size) $p$ (underlying prime, field characteristic) $r$ (size of Galois group) $P\Gamma L(2,q)$ Order of $P\Gamma L(2,q)$ (= $r(q^3 - q)$)
2 2 1 symmetric group:S3 6
3 3 1 symmetric group:S4 24
4 2 2 symmetric group:S5 120
5 5 1 symmetric group:S5 120
7 7 1 projective general linear group:PGL(2,7) 336
8 2 3 Ree group:Ree(3) 1512
9 3 2 automorphism group of alternating group:A6 1440
11 11 1 projective general linear group:PGL(2,11) 1320