General semilinear group
Contents
Definition
Suppose is a commutative unital ring and
is a natural number. The general semilinear group of degree
over
, denoted
, is defined as the group of all invertible
-semilinear transformations from a
-dimensional free module over
to itself.
It can also be described explicitly as an external semidirect product of the general linear group by the automorphism group of
as a commutative unital ring (acting entry-wise on the matrices), i.e.:
Special case of fields
Suppose is the prime subfield of
and suppose that
is a Galois extension over
(this is always true for
a finite field). Then,
and we can rewrite the group as:
where the action of on
is obtained by inducing the corresponding Galois automorphism on each matrix entry.
If is a prime power, we denote by
the group
where
is the (unique up to isomorphism) field of size
.
Arithmetic functions
For a finite field
We consider here a field of size
where
is the field characteristic, so
is a natural number.
The prime subfield is , and the extension
has degree
. The Galois group of the extension thus has size
. Note that the Galois group of the extension is always a cyclic group of order
and is generated by the Frobenius automorphism
.
We are interested in the group .
Function | Value | Explanation |
---|---|---|
order | ![]() |
order of semidirect product is product of orders: we multiply the order ![]() |
Particular cases
Finite cases
We consider a field of size where
is the underlying prime and field characteristic, and therefore
is the degree of the extension over the prime subfield and also the order of the Galois group.
Note that in the case , the general semilinear group coincides with the general linear group.
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order of ![]() |
---|---|---|---|---|---|
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 |