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 $\mathrm{\Gamma }A(n,K)$ or $A\mathrm{\Gamma }L(n,K)$, 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 $A\in GL(n,K)$, $b\in K^n$, $\sigma \in \mathrm{A}\mathrm{u}\mathrm{t}(K)$.
2. It is the semidirect product $GA(n,K)⋊\mathrm{A}\mathrm{u}\mathrm{t}(K)$ of the general affine group $GA(n,K)$ by the group $\mathrm{A}\mathrm{u}\mathrm{t}(K)$ 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 $K⋊\mathrm{\Gamma }L(n,K)$ of the additive group of $K$ by the general semilinear group $\mathrm{\Gamma }L(n,K)$ 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:

$K⋊\mathrm{\Gamma }L(n,K)=K⋊(GL(n,K)⋊\mathrm{A}\mathrm{u}\mathrm{t}(K))=(K⋊GL(n,K))⋊\mathrm{A}\mathrm{u}\mathrm{t}(K)=GA(n,K)⋊\mathrm{A}\mathrm{u}\mathrm{t}(K)$

Suppose $k$ is the prime subfield of $K$. Then, if $K$ is a Galois extension of $k$, $\mathrm{A}\mathrm{u}\mathrm{t}(K)$ is the Galois group $\mathrm{G}\mathrm{a}\mathrm{l}(K/k)$. This case always occurs if $K$ is a finite field.