Special linear group of degree two

For a field or commutative unital ring
The special linear group of degree two over a field $$k$$, or more generally over a commutative unital ring $$R$$, is defined as the group of $$2 \times 2$$ matrices with determinant $$1$$ under matrix multiplication, and entries over $$R$$. The group is denoted by $$SL(2,R)$$ or $$SL_2(R)$$.

When $$q$$ is a prime power, $$SL(2,q)$$ is the special linear group of degree two over the field (unique up to isomorphism) with $$q$$ elements.

The underlying set of the group is:

$$SL(2,R) := \left \{ \begin{pmatrix} a & b \\ c & d \\\end{pmatrix} \mid a,b,c,d \in R, ad - bc = 1 \right \}$$.

The group operation is given by:

$$\begin{pmatrix} a & b \\ c & d \\\end{pmatrix} \begin{pmatrix} a' & b' \\ c' & d' \\\end{pmatrix} = \begin{pmatrix} aa' + bc' & ab' + bd' \\ ca' + dc' & cb' + dd' \\\end{pmatrix}$$.

The identity element is:

$$\begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}$$.

The inverse map is given by:

$$\begin{pmatrix} a & b \\ c & d \\\end{pmatrix}^{-1} = \begin{pmatrix} d & -b \\ -c & a \\\end{pmatrix}$$

For a prime power
Let $$q$$ be a prime power. The special linear group $$SL(2,q)$$ is defined as $$SL(2,\mathbb{F}_q)$$, where $$\mathbb{F}_q$$ is the (unique up to isomorphism) field of size $$q$$.

Note that for a finite field, we have the following: special unitary group of degree two equals special linear group of degree two over a finite field. In other words, $$SL(2,q)$$ is isomorphic to $$SU(2,q)$$.

Over a finite field
Here, $$q$$ denotes the order of the finite field and the group we work with is $$SL(2,q)$$. $$p$$ is the characteristic of the field, i.e., it is the prime whose power $$q$$ is.

Over a finite field
As before, $$q$$ is the field size and $$p$$ is the characteristic of the field, so $$p$$ is a prime number and $$q$$ is a power of $$p$$.

Over a finite field
Here is a summary of the linear representation theory in characteristic zero (and all characteristics coprime to the order):

Here is a summary of the modular representation theory in characteristic $$p$$, where $$p$$ is the characteristic of the field over which we are taking the special linear group (so $$q$$ is a power of $$p$$):