Projective semilinear group
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).
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 .
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 .
|order||order of semidirect product is product of orders: we multiply the order of the Galois group with the order of the projective general linear group.|
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.
|(field size)||(underlying prime, field characteristic)||(degree of extension over prime subfield)||order of|
|7||7||1||2||projective general linear group:PGL(2,7)||336|
|9||3||2||2||automorphism group of alternating group:A6||1440|