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.

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.

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 $$(A - 1)$$.

