General semilinear group of degree one
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 .
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).