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).