# 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.