# General semilinear group of degree two

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## Definition

Let $K$ be a field. The general semilinear group of degree two over $K$, denoted $\Gamma L(2,K)$, is defined as the general semilinear group of degree two over $K$. Explicitly, it is the external semidirect product:

$\Gamma L (2,K) = GL(2,K) \rtimes \operatorname{Aut}(K)$

where $GL(2,K)$ denotes the general linear group of degree two and $\operatorname{Aut}(K)$ is the group of field automorphisms of $K$ acting entry-wise on the matrices.

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 $\operatorname{Aut}(K) = \operatorname{Gal}(K/k)$ (the Galois group) and we get:

$\Gamma L (2,K) = GL(2,K) \rtimes \operatorname{Gal}(K/k)$