# Changes

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

, 22:50, 1 March 2012
Particular cases
{| class="sortable" border="1"
! $q</math><br> (field size) !! [itex]p</math> <br> (underlying prime, field characteristic) !! [itex]r = \log_pq$ !! 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 ||
|}
| $A</matH> is the identity, [itex]v \ne 0$ || $\{ 1,1 \}$ || $(x - 1)^2$ || $x - 1$ || $q^2 - 1$ || 1 || <matH>q^2 - 1[/itex] || Yes || Yes
|-
| $A$ is diagonalizable over $\mathbb{F}_q$ with equal diagonal entries not equal to 1, hence a scalar. The value of <matH>v[/itex] 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 $v</matH> does not affect the conjugacy class. || Pair of conjugate elements of [itex]\mathbb{F}_{q^2}$ || $x^2 - ax + b$, 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
| $A$ diagonalizable over $\mathbb{F}_q$ with ''distinct'' diagonal entries, one of which is 1, $v$ is not in the image of $A - 1</matH> || [itex]1,\mu$, $\mu \in \mathbb{F}_q^\ast \setminus \{ 1 \}$ || $x^2 - (\mu + 1)x + \mu$ || 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 !! $q^2 + q - 1$ !! $q^2(q^2 - 1)(q^2 - q)$ !! !!
|}
<section end="conjugacy class structure"/>