# General semilinear group of degree one

## Definition

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

where is the multiplicative group of , and denotes the group of field automorphisms of .

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

If is a finite field of size , this group is written as .

## Particular cases

### For a finite field

Suppose is a finite field of size , where is a prime power with underlying prime , so that for a positive integer . is the characteristic of . In this case, is cyclic of order (see multiplicative group of a finite field is cyclic) and is cyclic of order (generated by the Frobenius map ).

Thus, is a metacyclic group of order with presentation:

(here denotes the identity element).