General semiaffine group

Definition
Suppose $$K$$ is a field and $$n$$ is a natural number. The general semiaffine group of degree $$n$$ over $$K$$, denoted $$\Gamma A(n,K)$$ or $$A \Gamma L(n,K)$$, is defined in either of these equivalent ways:


 * 1) It is the group of all maps from $$K^n$$ to $$K^n$$ of the form $$v \mapsto A \sigma(v) + b$$ where $$A \in GL(n,K)$$, $$b \in K^n$$, $$\sigma \in \operatorname{Aut}(K)$$.
 * 2) It is the semidirect product $$GA(n,K) \rtimes \operatorname{Aut}(K)$$ of the general affine group $$GA(n,K)$$ by the group $$\operatorname{Aut}(K)$$ of field automorphisms of $$K$$, 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.
 * 3) It is the semidirect product $$K \rtimes \Gamma L(n,K)$$ of the additive group of $$K$$ by the general semilinear group $$\Gamma L(n,K)$$ 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:

$$K \rtimes \Gamma L(n,K) = K \rtimes (GL(n,K) \rtimes \operatorname{Aut}(K)) = (K \rtimes GL(n,K)) \rtimes \operatorname{Aut}(K) = GA(n,K) \rtimes \operatorname{Aut}(K)$$

Suppose $$k$$ is the prime subfield of $$K$$. Then, if $$K$$ is a Galois extension of $$k$$, $$\operatorname{Aut}(K)$$ is the Galois group $$\operatorname{Gal}(K/k)$$. This case always occurs if $$K$$ is a finite field.