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).