# Difference between revisions of "Element structure of general affine group of degree two over a finite field"

## Contents

This article gives specific information, namely, element structure, about a family of groups, namely: general affine group of degree two.
View element structure of group families | View other specific information about general affine group of degree two

This article gives the element structure of the general affine group of degree two over a finite field. Similar structure works over an infinite field or a field of infinite characteristic, with suitable modification. For more on that, see element structure of general affine group of degree two over a field.

The discussion here builds upon the discussion of element structure of general linear group of degree two over a finite field.

## Summary

Item Value
order $q^2(q^2 - 1)(q^2 - q) = q^3(q + 1)(q - 1)^2$
exponent  ?
number of conjugacy classes $q^2 + q - 1$

## Particular cases

$q$ (field size) $p$ (underlying prime, field characteristic) general affine group $GL(2,q)$ order of the group (= $q^2(q^2 - 1)(q^2 - q)$) number of conjugacy classes (= $q^2 + q - 1$) element structure page
2 2 symmetric group:S4 24 5 element structure of symmetric group:S4
3 3 general affine group:GA(2,3) 432 11 element structure of general affine group:GA(2,3)
4 2 general affine group:GA(2,4) 2880 19
5 5 general affine group:GA(2,5) 12000 29

## Conjugacy class structure

There is a total of $q^2(q^2 - 1)(q^2 - q) = q^3(q - 1)^2(q + 1)$ elements, and there are $q^2 + q - 1$ conjugacy classes of elements.

The conjugacy class structure is closely related to that of $GL(2,q)$ -- see Element structure of general linear group of degree two over a finite field#Conjugacy class structure.

We describe a generic element of $GA(2,q)$ in the form:

$x \mapsto Ax + v$

where $A \in GL(2,q)$ is the dilation component and $v \in (\mathbb{F}_q)^2$ is the translation component.

Consider the quotient mapping $GA(2,q) \to GL(2,q)$, which sends the generic element to $A$. Under this mapping, the following is true:

• For those conjugacy classes of $GL(2,q)$ comprising elements that do not have 1 as an eigenvalue, the full inverse image of the conjugacy class is a single conjugacy class in $GA(2,q)$. In other words, the translation component does not matter.
• For those conjugacy classes of $GL(2,q)$ comprising elements that do have 1 as an eigenvalue, the conjugacy class splits into two depending on whether $v$ is in the image of $(A - 1)$.
Nature of conjugacy class Eigenvalues Characteristic polynomial of $A$ Minimal polynomial of $A$ Size of conjugacy class Number of such conjugacy classes Total number of elements Is $A$ semisimple? Is $A$ diagonalizable over $\mathbb{F}_q$?
$A$ is the identity, $v = 0$ $\{ 1,1 \}$ $(x - 1)^2$ $x - 1$ 1 1 1 Yes Yes
$A$ is the identity, $v \ne 0$ $\{ 1,1 \}$ $(x - 1)^2$ $x - 1$ $q^2 - 1$ 1 $q^2 - 1$ Yes Yes
$A$ is diagonalizable over $\mathbb{F}_q$ with equal diagonal entries not equal to 1, hence a scalar. The value of $v$ does not affect the conjugacy class. $\{a,a \}$ where $a \in \mathbb{F}_q^\ast \setminus \{ 1 \}$ $(x - a)^2$ where $a \in \mathbb{F}_q^\ast$ $x - a$ where $a \in \mathbb{F}_q^\ast$ $q^2$ $q - 2$ $q^2(q - 2)$ Yes Yes
$A$ is diagonalizable over $\mathbb{F}_{q^2}$, not over $\mathbb{F}_q$. Must necessarily have no repeated eigenvalues. The value of $v$ does not affect the conjugacy class. Pair of conjugate elements of $\mathbb{F}_{q^2}$ $x^2 - ax + b$, irreducible Same as characteristic polynomial $q^3(q - 1)$ $q(q - 1)/2 = (q^2 - q)/2$ $q^4(q-1)^2/2$ Yes No
$A$ has Jordan block of size two, with repeated eigenvalue equal to 1, $v = 0$ $\{ 1, 1 \}$ $(x - 1)^2$ Same as characteristic polynomial $q^2 - 1$ 1 $q^2 - 1$ No No
$A$ has Jordan block of size two, with repeated eigenvalue equal to 1, $v \ne 0$ $\{ 1, 1 \}$ $(x - 1)^2$ Same as characteristic polynomial $(q^2 - 1)^2$ 1 $(q^2 - 1)^2$ No No
$A$ has Jordan block of size two, with repeated eigenvalue not equal to 1 $a$ (multiplicity two) where $a \in \mathbb{F}_q^\ast \setminus \{ 1 \}$ $(x - a)^2$ where $a \in \mathbb{F}_q^\ast \setminus \{ 1 \}$ Same as characteristic polynomial $q^2(q^2 - 1)$ $q - 2$ $q^2(q^2 - 1)(q - 2)$ No No
$A$ diagonalizable over $\mathbb{F}_q$ with distinct diagonal entries, one of which is 1, $v = 0$ $1,\mu$, $\mu \in \mathbb{F}_q^\ast \setminus \{ 1 \}$ $x^2 - (\mu + 1)x + \mu$ Same as characteristic polynomial $q(q + 1)$ $q - 2$ $q(q+1)(q-2)$ Yes Yes
$A$ diagonalizable over $\mathbb{F}_q$ with distinct diagonal entries, one of which is 1, $v \ne 0$ $1,\mu$, $\mu \in \mathbb{F}_q^\ast \setminus \{ 1 \}$ $x^2 - (\mu + 1)x + \mu$ Same as characteristic polynomial $q(q + 1)(q^2 - 1)$ $q - 2$ $q(q+1)^2(q - 1)(q-2)$ Yes Yes
$A$ diagonalizable over $\mathbb{F}_q$ with distinct diagonal entries, neither of which is 1 $\lambda, \mu$ (interchangeable) distinct elements of $\mathbb{F}_q^\ast$ $x^2 - (\lambda + \mu)x + \lambda \mu$ Same as characteristic polynomial $q^3(q+1)$ $(q - 2)(q - 3)/2$ $q^3(q+1)(q-2)(q-3)/2$ Yes Yes
Total NA NA NA NA $q^2 + q - 1$ $q^2(q^2 - 1)(q^2 - q)$