General linear group of degree two

Definition

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

For a prime power $q$, $GL(2,q)$ or $G{L}_2(q)$ denotes the general linear group of degree two over the field (unique up to isomorphism) with $q$ elements.

Arithmetic functions

Here, $q$ denotes the order of the finite field and the group we work with is $GL(2,q)$.

Function	Value	Explanation
order	$q^3-q=q(q-1)(q+1)$	$q^2-1$ options for first row, $q^2-q$ options for second row.
exponent	$q^3-q=q(q-1)(q+1)$	There is an element of order $q^2-1$ and an element of order $q$.
number of conjugacy classes	$q^2-1$	There are $q(q-1)$ conjugacy classes of semisimple matrices and $q-1$ conjugacy classes of matrices with repeated eigenvalues.

Group properties

Property	Satisfied	Explanation
Abelian group	No	The matrices $\left(\begin{array}{cc}1& 1\\ 0& 1\\ \end{array}\right)$ and $\left(\begin{array}{cc}1& 0\\ 1& 1\\ \end{array}\right)$ don't commute.
Nilpotent group	No	$PSL(2,q)$ is simple for $q\ge 4$, and we can check the cases $q=2,3$ separately.
Solvable group	Yes if $q=2,3$, no otherwise.	$PSL(2,q)$ is simple for $q\ge 4$.
Supersolvable group	Yes if $q-2$, no otherwise.	$PSL(2,q)$ is simple for $q\ge 4$, and we can check the cases $q=2,3$ separately.