General semilinear group

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Suppose R is a commutative unital ring and n is a natural number. The general semilinear group of degree n over R, denoted \Gamma L(n,R), is defined as the group of all invertible R-semilinear transformations from a n-dimensional free module over R to itself.

It can also be described explicitly as an external semidirect product of the general linear group GL(n,R) by the automorphism group of R as a commutative unital ring (acting entry-wise on the matrices), i.e.:

\Gamma L(n,R) = GL(n,R) \rtimes \operatorname{Aut}(R)

Special case of fields

Suppose k is the prime subfield of K and suppose that K is a Galois extension over k (this is always true for K a finite field). Then, \operatorname{Aut}(K) = \operatorname{Gal}(K/k) and we can rewrite the group as:

\Gamma L(n,K) = GL(n,K) \rtimes \operatorname{Gal}(K/k)

where the action of \operatorname{Gal}(K/k) on GL(n,K) is obtained by inducing the corresponding Galois automorphism on each matrix entry.

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

Arithmetic functions

For a finite field

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 \Gamma L(n,q).

Function Value Explanation
order rq^{\binom{n}{2}}\prod_{i=1}^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 general linear group.

Particular cases

Finite 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 general semilinear group coincides with the general linear 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 cyclic group:Z2 1
4 2 2 1 symmetric group:S3 6
5 5 1 1 cyclic group:Z4 4
7 7 1 1 cyclic group:Z6 6
8 2 3 1 semidirect product of Z7 and Z3 21
2 2 1 2 symmetric group:S3 6
3 3 1 2 general linear group:GL(2,3) 48
4 2 2 2 general semilinear group:GammaL(2,4) 360
5 5 1 2 general linear group:GL(2,5) 480