Outer semilinear group

Definition

Suppose $K$ is a field and $n$ is a natural number. The outer semilinear group of degree $n$ over $K$, denoted $O\Gamma L(n,K)$, is defined as the external semidirect product:

$GL(n,K) \rtimes (\operatorname{Aut}(K) \times \mathbb{Z}/2\mathbb{Z})$

where $\operatorname{Aut}(K)$ acts coordinate-wise on the matrix entries by automorphisms and the non-identity element of $\mathbb{Z}/2\mathbb{Z}$ acts by the transpose-inverse map. Note that the two actions commute with each other, so we can combine these to get the action of the external direct product.

In the case that $K$ is a Galois extension of its prime subfield $k$ (note that this is always true when $K$ is a finite field), $\operatorname{Aut}(K) = \operatorname{Gal}(K/k)$, so the group can also be written as:

$GL(n,K) \rtimes (\operatorname{Gal}(K/k) \times \mathbb{Z}/2\mathbb{Z})$