General semiaffine group of degree one

Definition
The general semiaffine group of degree one over a field $$K$$, denoted $$\Gamma A(1,K)$$ or $$A\Gamma L(1,K)$$, is the general semiaffine group of degree one over $$K$$. It can be defined explicitly as the group of all permutations of $$K$$ that can be expressed in the form:

$$x \mapsto a \sigma(x) + b, a \in K^\ast, b \in K, \sigma \in \operatorname{Aut}(K)$$

It can be viewed as an iterated semidirect product:

$$(K \rtimes K^\ast) \rtimes \operatorname{Aut}(K) = K \rtimes (K^\ast \rtimes \operatorname{Aut}(K))$$

The left parenthesized expression shows that the group can be viewed as a semidirect product with base the general affine group of degree one and acting group the Galois group:

$$\Gamma A(1,K) = GA(1,K) \rtimes \operatorname{Aut}(K)$$

The right parenthesized expression shows that the group can be viewed as a semidirect product with base the additive group of the field and acting group the general semilinear group of degree one:

$$\Gamma A(1,K) = K \rtimes \Gamma L(1,K)$$

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

Alternative definition as automorphisms of a polynomial ring
The group $$\Gamma A(1,K)$$ can also be defined as the group of all ring automorphisms of the polynomial ring $$K[x]$$. The subgroup $$\operatorname{Aut}_K(K[x])$$ of those automorphisms that fix the base field $$K$$ can be identified with the general affine group of degree one $$GA(1,K)$$.