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

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)	$r=\log _{p}q$	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, $v\neq 0$	$\{1,1\}$	$(x-1)^{2}$	$x-1$	$q^{2}-1$	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	$q^{2}+q-1$	$q^{2}(q^{2}-1)(q^{2}-q)$