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

This article gives the element structure of the general linear group of degree two over a finite field. Similar structure works over an infinite field or a field of infinite characteristic, with suitable reinterpretation and modification. For more on that, see element structure of general linear group of degree two over a field and element structure of general linear group over a field.

See also element structure of projective general linear group of degree two, element structure of special linear group of degree two, element structure of projective special linear group of degree two.

We denote the order (or size) of the field by $$q$$ and the characteristic of the field by $$p$$. $$q$$ is a power of $$p$$.

Conjugacy class structure


There is a total of $$q(q+1)(q - 1)^2 = q^4 - q^3 - q^2 + q$$ elements, and there are $$q^2 - 1 = (q - 1)(q +1)$$ conjugacy classes of elements.

For background on how this conjugacy class structure can be obtained and also generalized to general linear groups of degree three or more, refer to conjugacy class size formula in general linear group over finite field.



Central elements
The center is a subgroup of order $$q - 1$$, and its elements are diagonal matrices of the form:

$$\{ \begin{pmatrix} a & 0 \\ 0 & a \\\end{pmatrix} : a \in \mathbb{F}_q^\ast \}$$

The subgroup is isomorphic to the multiplicative group of the field of $$q$$ elements, and since multiplicative group of finite field is cyclic, it is a cyclic group of order $$q - 1$$.

Real and rational conjugacy
We have the following:

Order information
For a given $$d$$ dividing $$q - 1$$:

Elements diagonalizable over $$\mathbb{F}_{q^2}$$ but not $$\mathbb{F}_q$$
These elements have pairs of distinct eigenvalues over $$\mathbb{F}_{q^2}$$ that are conjugate over $$\mathbb{F}_q$$. The unique non-identity automorphism of $$\mathbb{F}_{q^2}$$ over $$\mathbb{F}_q$$ is the map $$x \mapsto x^q$$, so these two elements are $$q^{th}$$ powers of each other, i.e., if one of them is $$\alpha$$, the other one is $$\alpha^q$$.

The conjugacy class is parameterized by the pair $$\{ \alpha, \alpha^q \}$$. Here is more information:

Here is information on the collection of all such conjugacy classes:

Order information
A number $$d$$ occurs as the order of an element in one of these conjugacy classes if $$d | q^2 - 1$$ but $$d$$ does not divide $$q - 1$$. More information below:

Elements diagonalizable over $$\mathbb{F}_q$$ with distinct diagonal entries
Note that we are including here only invertible elements, hence both eigenvalues must be in $$\mathbb{F}_q^\ast$$.

Each such conjugacy class is specified by an unordered pair of distinct elements of $$\mathbb{F}_q^\ast$$, say $$\lambda, \mu$$.

Here is combined information for all conjugacy classes:

Order information
A natural number $$d$$ occurs as the order of an element in one of these conjugacy classes if and only if $$d$$ divides $$q - 1$$ but is greater than $$1$$. For such a value of $$d$$, we have:

Elements with Jordan block of size two
These are elements conjugate to elements of the form:

$$\{ \begin{pmatrix} a & 1 \\ 0 & a \\\end{pmatrix}: a \in \mathbb{F}_q^\ast \}$$

Each conjugacy class is parameterized by the value of $$a$$.

Here is the information on the collection of all conjugacy classes:

Order information
The possible orders are $$dp$$, where $$d$$ divides $$q - 1$$ and $$p$$ is the underlying prime (the characteristic) whose power is $$q$$. For each order $$dp$$, the number of conjugacy classes is $$\varphi(d)$$, the Euler totient function.