General semilinear group of degree one

Definition

Let $K$ be a field. The general semilinear group of degree one over $K$, denoted $\Gamma L(1,K)$, is defined as the general semilinear group of degree one over $K$. Explicitly, it is the external semidirect product:

$\Gamma L (1,K) = GL(1,K) \rtimes \operatorname{Aut}(K) = K^\ast \rtimes \operatorname{Aut}(K)$

where $GL(1,K) = K^\ast$ is the multiplicative group of $K$, and $\operatorname{Aut}(K)$ denotes the group of field automorphisms of $K$.

If $k$ is the prime subfield of $K$, and $K$ is a Galois extension of $k$ (note that this case always occurs for $K$ a finite field), then $\operatorname{Aut}(K) = \operatorname{Gal}(K/k)$ and we get:

$\Gamma L (1,K) = GL(1,K) \rtimes \operatorname{Gal}(K/k) = K^\ast \rtimes \operatorname{Gal}(K/k)$

If $K$ is a finite field of size $q$, this group is written as $\Gamma L(1,q)$.

Particular cases

For a finite field

Suppose $K$ is a finite field of size $q$, where $q$ is a prime power with underlying prime $p$, so that $q = p^r$ for a positive integer $r$. $p$ is the characteristic of $K$. In this case, $K^\ast$ is cyclic of order $q - 1$ (see multiplicative group of a finite field is cyclic) and $\operatorname{Gal}(K/k)$ is cyclic of order $r$ (generated by the Frobenius map $a \mapsto a^p$).

Thus, $\Gamma L(1,K)$ is a metacyclic group of order $r(q - 1)$ with presentation:

$\langle a,x \mid a^q = a, x^r = e, xax^{-1} = a^p \rangle$

(here $e$ denotes the identity element).