General affine group of degree one: Difference between revisions

Latest revision as of 20:24, 17 November 2023

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 $G A (1, K)$ or $A G L (1, K)$ , and can be written as:

$G A (1, K) = K ⋊ K^{*}$

Alternative definition as automorphisms of a polynomial ring

For a field $K$ , the general affine group of degree one $G A (1, K)$ can be defined as the group $A_{u t K} (K [x])$ .

Note that this definition does not extend to general affine groups of higher degree. For $n > 1$ , $G A (n, K)$ naturally sits as a subgroup inside $A_{u t K} (K [x_{1}, x_{2}, \dots, x_{n}])$ 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 $G A (1, q)$ is defined as follows. Let $F_{q}$ denote the field with $q$ elements. Then $G A (1, q)$ is the semidirect product of the additive group of $F_{q}$ 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)	$G A (1, q)$	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 $F_{q}$ , which has order $q$ , and the multiplicative group of $F_{q}$ , 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 $F_{q}$ 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.

Linear representation theory

Further information: Linear representation theory of general affine group of degree one over a finite field

@@ Line 1: / Line 1: @@
 ==Definition==
+===For a field===
+For a field <math>K</math>, the general affine group of degree one over <math>K</math> is defined as the [[general affine group]] over <math>K</math> of degree one. Equivalently, it is the [[external semidirect product]] of the additive group of <math>K</math> by the multiplicative group of <math>K</math>, where the latter acts naturally on the former by field multiplication. Explicitly, it is denoted <math>GA(1,K)</math> or <math>AGL(1,K)</math>, and can be written as:
+<math>GA(1,K) = K \rtimes K^\ast</math>
+===Alternative definition as automorphisms of a polynomial ring===
+For a field <math>K</math>, the general affine group of degree one <math>GA(1,K)</math> can be defined as the group <math>\operatorname{Aut}_K(K[x])</math>.
+Note that this definition does not extend to [[general affine group]]s of higher degree. For <math>n > 1</math>, <math>GA(n,K)</math> naturally sits as a subgroup inside <math>\operatorname{Aut}_K(K[x_1,x_2,\dots,x_n])</math> but is ''not'' the whole automorphism group.
+===For a finite number===
 Let <math>p</math> be a [[prime number]] and <math>q = p^r</math> be a power of <math>p</math>. The '''general affine group''' or '''collineation group''' <math>GA(1,q)</math> is defined as follows. Let <math>\mathbb{F}_q</math> denote the field with <math>q</math> elements. Then <math>GA(1,q)</math> is the semidirect product of the additive group of <math>\mathbb{F}_q</math> with its multiplicative group.
 Equivalently it is the [[general affine group]] of degree <math>1</math> over the field of <math>q</math> elements.
+==Particular cases==
+{| class="sortable" border="1"
+! <math>q</math> (field size) !! <math>p</math> (underlying prime, field characteristic) !! <math>GA(1,q)</math> !! 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, <math>q</math> is the size of the field and <matH>p</math> is the underlying prime (the characteristic of the field). We have <math>q = p^r</math> where <math>r</math> is a positive integer.
+{| class="sortable" border="1"
+! Function !! Value !! Explanation
+|-
+| [[order of a group|order]] || <math>q(q - 1) = q^2 - q</math> || [[order of semidirect product is product of orders]]: The group is a semidirect product of the additive group of <math>\mathbb{F}_q</math>, which has order <math>q</math>, and the multiplicative group of <math>\mathbb{F}_q</math>, which has order <math>q - 1</math> (because it comprises all the non-identity elements).
+|-
+| [[exponent of a group|exponent]] || <math>p(q - 1)</math> || Non-identity elements in the additive group have order <math>p</math> and all elements outside it have order <math>q - 1</math>.
+|-
+| [[derived length]] || 2 || The [[derived subgroup]] is the additive group. The exception is the case <math>q = 2</math>, 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 <math>q > 2</math>, we can find two maximal subgroups of order <math>q - 1</math> with trivial intersection. Note that this also follows from it being a [[Frobenius group]].
+|}
+===Arithmetic functions of a counting nature===
+{| class="sortable" border="1"
+! Function !! Value !! Explanation
+|-
+| [[number of conjugacy classes]] ||<math>q</math> || identity element, one conjugacy class of non-identity elements of additive group, <math>q - 2</math> conjugacy classes, one for each non-identity coset of the additive group.
+|}
 ==Group properties==
-===Frobenius group===
+{| class="sortable" border="1"
+! Property !! Satisfied? !! Explanation !! Corollary properties satisfied/dissatisfied
-The additive subgroup <math>\mathbb{F}_q</math> is a [[Frobenius kernel]] and the multiplicative subgroup is a [[Frobenius complement]]. (note: the case <math>q = 2</math> is an exception, where it fails to be a Frobenius group on account of the multiplicative group being trivial).
+|-
+| [[satisfies property::Frobenius group]] || Yes || The additive subgroup <math>\mathbb{F}_q</math> is a [[Frobenius kernel]] and the multiplicative subgroup is a [[Frobenius complement]]. (note: the case <math>q = 2</math> is an exception, where it fails to be a Frobenius group on account of the multiplicative group being trivial). ||
+|-
+| [[satisfies property::Camina group]] || Yes || The derived subgroup is the additive group, and every coset of that forms a conjugacy class. ||
+|-
+| [[dissatisfies property::abelian group]] || No || Except the <math>q = 2</math> case, where we get [[cyclic group:Z2]] ||
+|-
+| [[dissatisfies property::nilpotent group]] || No || Except the <math>q = 2</math> case, where we get [[cyclic group:Z2]] ||
+|-
+| [[satisfies property::metabelian group]] || Yes || The [[derived subgroup]] is the additive group of the field (when <math>q > 2</math>). || Satisfies: [[satisfies property::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 <math>q</math> is a [[prime number]] rather than a strict prime power.
+|}
-===Solvable group===
+==Linear representation theory==
-The group is solvable of solvable length two. Specifically, its commutator subgroup is precisely the additive group of the field.
+{{further|[[Linear representation theory of general affine group of degree one over a finite field]]}}