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.