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