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)