General affine group of degree one

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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.