General semilinear group of degree one

Definition

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

$\mathrm{\Gamma }L(1,K)=GL(1,K)⋊\mathrm{A}\mathrm{u}\mathrm{t}(K)=K^{\ast }⋊\mathrm{A}\mathrm{u}\mathrm{t}(K)$

where $GL(1,K)=K^{\ast }$ is the multiplicative group of $K$, and $\mathrm{A}\mathrm{u}\mathrm{t}(K)$ 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 $\mathrm{A}\mathrm{u}\mathrm{t}(K)=\mathrm{G}\mathrm{a}\mathrm{l}(K/k)$ and we get:

$\mathrm{\Gamma }L(1,K)=GL(1,K)⋊\mathrm{G}\mathrm{a}\mathrm{l}(K/k)=K^{\ast }⋊\mathrm{G}\mathrm{a}\mathrm{l}(K/k)$

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

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 $\mathrm{G}\mathrm{a}\mathrm{l}(K/k)$ is cyclic of order $r$ (generated by the Frobenius map $a\mapsto a^p$).

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

$⟨a,x\mid a^q=a,x^r=e,xa{x}^{-1}=a^p⟩$

(here $e$ denotes the identity element).