Projective semilinear group

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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