Projective semilinear group
Definition
Let be a field and
be a natural number.
The projective semilinear group or projective general semilinear group is defined as the quotient group of the general semilinear group
by the subgroup corresponding to scalar linear transformations, i.e., the subgroup corresponding to the center of
. (Note that the subgroup by which we are quotienting is not in the center of
in general).
It can also be defined as an external semidirect product of the projective general linear group by a Galois group:
where is the prime subfield of
.
For a prime power , we denote by
the group
where
is the (unique up to isomorphism) field of size
.
Arithmetic functions
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 | Similar groups | Explanation |
---|---|---|---|
order | ![]() |
order of semidirect product is product of orders: we multiply the order ![]() |
Particular 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 projective general semilinear group coincides with the projective general linear group.
In the case , the projective semilinear group is just the Galois group, which is cyclic of order
.
Finally, if , we get a trivial group.
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order of ![]() |
---|---|---|---|---|---|
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 |