Element structure of general affine group of degree two over a finite field

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)	$r = \log_pq$	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	1	symmetric group:S4	24	5	element structure of symmetric group:S4
3	3	1	general affine group:GA(2,3)	432	11	element structure of general affine group:GA(2,3)
4	2	2	general affine group:GA(2,4)	2880	19
5	5	1	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 $a \in \mathbb{F}_q^\ast \setminus \{ 1 \}$	$x - a$ where $a \in \mathbb{F}_q^\ast \setminus \{ 1 \}$	$q^2$	$q - 2$	$q^2(q - 2)$	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, $v$ is in the image of $A - 1$	$\{ 1, 1 \}$	$(x - 1)^2$	Same as characteristic polynomial	$q(q^2 - 1)$	1	$q(q^2 - 1)$	No	No
$A$ has Jordan block of size two, with repeated eigenvalue equal to 1, $v$ is not in the image of $A - 1$	$\{ 1, 1 \}$	$(x - 1)^2$	Same as characteristic polynomial	$(q^2 - 1)(q^2 - q)$	1	$(q^2 - 1)(q^2 - q)$	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, $v$ is in the image of $A - 1$	$1,\mu$ , $\mu \in \mathbb{F}_q^\ast \setminus \{ 1 \}$	$x^2 - (\mu + 1)x + \mu$	Same as characteristic polynomial	$q^2(q + 1)$	$q - 2$	$q^2(q+1)(q-2)$	Yes	Yes
$A$ diagonalizable over $\mathbb{F}_q$ with distinct diagonal entries, one of which is 1, $v$ is not in the image of $A - 1$	$1,\mu$ , $\mu \in \mathbb{F}_q^\ast \setminus \{ 1 \}$	$x^2 - (\mu + 1)x + \mu$	Same as characteristic polynomial	$q(q + 1)(q^2 - q)$	$q - 2$	$q^2(q+1)(q - 1)(q-2)$	Yes	Yes
$A$ diagonalizable over $\mathbb{F}_q$ with distinct diagonal entries, neither of which is 1	$\lambda, \mu$ (interchangeable) distinct elements of $\mathbb{F}_q^\ast$ , neither equal to 1	$x^2 - (\lambda + \mu)x + \lambda \mu$	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	$q^2 + q - 1$	$q^2(q^2 - 1)(q^2 - q)$