# General semilinear group of degree two

## Definition

Let be a field. The **general semilinear group of degree two** over , denoted , is defined as the general semilinear group of degree two over . Explicitly, it is the external semidirect product:

where denotes the general linear group of degree two and is the group of field automorphisms of acting entry-wise on the matrices.

If is the prime subfield of , and is a Galois extension of (note that this case always occurs for a finite field), then (the Galois group) and we get: