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.