General semiaffine group
Suppose is a field and is a natural number. The general semiaffine group of degree over , denoted or , is defined in either of these equivalent ways:
- It is the group of all maps from to of the form where , , .
- It is the semidirect product of the general affine group by the group of field automorphisms of , where the latter acts on the former by making the automorphism act coordinate-wise on the entries of the matrix and on the entries of the translation vector.
- It is the semidirect product of the additive group of by the general semilinear group with the natural action of the latter on the former.
We can think of the group as an iterated semidirect product that can be associated two ways:
Suppose is the prime subfield of . Then, if is a Galois extension of , is the Galois group . This case always occurs if is a finite field.