# Changes

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

, 23:40, 22 February 2012
Conjugacy class structure
| $A</matH> is the identity, [itex]v \ne 0$ || $\{ 1,1 \}$ || $(x - 1)^2$ || $x - 1$ || $q^2 - 1$ || 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