General linear group of degree two

For a unital ring
The general linear group of degree two over a unital ring $$R$$ is defined as the group, under matrix multiplication, of invertible $$2 \times 2$$ matrices with entries in $$R$$. It is denoted $$GL(2,R)$$.

For a commutative unital ring
When $$R$$ is a commutative unital ring, a $$2 \times 2$$ matrix over $$R$$ being invertible is equivalent to its determinant being an invertible element of $$R$$, so the general linear group $$GL(2,R)$$ is defined as the following group of matrices under matrix multiplication:

$$GL(2,R) := \left \{ \begin{pmatrix} a & b \\ c & d \\\end{pmatrix} \mid a,b,c,d \in R, ad - bc \mbox{ is an invertible element of } R \right \}$$

For a field
For a field $$K$$, an element is invertible iff it is nonzero, so the general linear group $$GL(2,K)$$ is defined as the following group of matrices under matrix multiplication:

$$GL(2,K) := \left \{ \begin{pmatrix} a & b \\ c & d \\\end{pmatrix} \mid a,b,c,d \in K, ad - bc \ne 0 \right \}$$

For a prime power
For a prime power $$q$$, $$GL(2,q)$$ or $$GL_2(q)$$ denotes the general linear group of degree two over the finite field (unique up to isomorphism) with $$q$$ elements. This is a field of characteristic $$p$$, where $$p$$ is the prime number whose power is $$q$$.

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