General affine group of degree one

Definition

For a field

For a field $K$, the general affine group of degree one over $K$ is defined as the general affine group over $K$ of degree one. Equivalently, it is the external semidirect product of the additive group of $K$ by the multiplicative group of $K$, where the latter acts naturally on the former by field multiplication. Explicitly, it is denoted $GA(1,K)$ or $AGL(1,K)$, and can be written as:

$GA(1,K) = K \rtimes K^\ast$

Alternative definition as automorphisms of a polynomial ring

For a field $K$, the general affine group of degree one $GA(1,K)$ can be defined as the group $\operatorname{Aut}_K(K[x])$.

Note that this definition does not extend to general affine groups of higher degree. For $n > 1$, $GA(n,K)$ naturally sits as a subgroup inside $\operatorname{Aut}_K(K[x_1,x_2,\dots,x_n])$ but is not the whole automorphism group.

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 all elements outside it 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.

Arithmetic functions of a counting nature

Function	Value	Explanation
number of conjugacy classes	$q$	identity element, one conjugacy class of non-identity elements of additive group, $q - 2$ conjugacy classes, one for each non-identity coset of the additive 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).
Camina group	Yes	The derived subgroup is the additive group, and every coset of that forms a conjugacy class.
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.