General affine group of degree one
Let be a prime number and be a power of . The general affine group or collineation group is defined as follows. Let denote the field with elements. Then is the semidirect product of the additive group of with its multiplicative group.
Equivalently it is the general affine group of order over the field of elements.
The additive subgroup is a Frobenius kernel and the multiplicative subgroup is a Frobenius complement. (note: the case is an exception, where it fails to be a Frobenius group on account of the multiplicative group being trivial).
The group is solvable of solvable length two. Specifically, its commutator subgroup is precisely the additive group of the field.