Projective semilinear group of degree two

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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