General affine group of degree one

Definition

For a field

For a field $F$, the general affine group of degree one over $F$ is defined as the general affine group over $F$ of degree one. Equivalently, it is the external semidirect product of the additive group of $F$ by the multiplicative group of $F$, where the latter acts naturally on the former by field multiplication.

For a finite number

Let $p$ be a prime number and $q=p^r$ be a power of $p$. The general affine group or collineation group $GA(1,q)$ is defined as follows. Let ${\mathbb{F}}_q$ denote the field with $q$ elements. Then $GA(1,q)$ is the semidirect product of the additive group of ${\mathbb{F}}_q$ with its multiplicative group.

Equivalently it is the general affine group of degree $1$ over the field of $q$ elements.

Particular cases

$q$ (field size)	$p$ (underlying prime, field characteristic)	$GA(1,q)$	Order	Second part of GAP ID
2	2	cyclic group:Z2	2	1
3	3	symmetric group:S3	6	1
4	2	alternating group:A4	12	3
5	5	general affine group:GA(1,5)	20	3
7	7	general affine group:GA(1,7)	42	1
8	2	general affine group:GA(1,8)	56	11
9	3	general affine group:GA(1,9)	72	39

Arithmetic functions

Below, $q$ is the size of the field and $p$ is the underlying prime (the characteristic of the field). We have $q=p^r$ where $r$ is a positive integer.

Function	Value	Explanation
order	$q(q-1)=q^2-q$	order of semidirect product is product of orders: The group is a semidirect product of the additive group of ${\mathbb{F}}_q$, which has order $q$, and the multiplicative group of ${\mathbb{F}}_q$, which has order $q-1$ (because it comprises all the non-identity elements).
exponent	$p(q-1)$	Non-identity elements in the additive group have order $p$ and elements in the multiplicative group have order $q-1$.
derived length	2	The derived subgroup is the additive group. The exception is the case $q=2$, where the group is abelian and has derived length 1.
Fitting length	2	The Fitting subgroup is the additive group of the field, and the quotient is an abelian group.
Frattini length	1	For $q>2$, we can find two maximal subgroups of order $q-1$ with trivial intersection. Note that this also follows from it being a Frobenius group.

Group properties

Property	Satisfied?	Explanation	Corollary properties satisfied/dissatisfied
Frobenius group	Yes	The additive subgroup ${\mathbb{F}}_q$ is a Frobenius kernel and the multiplicative subgroup is a Frobenius complement. (note: the case $q=2$ is an exception, where it fails to be a Frobenius group on account of the multiplicative group being trivial).
abelian group	No	Except the $q=2$ case, where we get cyclic group:Z2
nilpotent group	No	Except the $q=2$ case, where we get cyclic group:Z2
metabelian group	Yes	The derived subgroup is the additive group of the field (when $q>2$).	Satisfies: solvable group
supersolvable group	Sometimes	The group is supersolvable if and only if the field is a prime field, i.e., if and only if $q$ is a prime number rather than a strict prime power.