# General linear group of degree two

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## 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 $GL_2(q)$ 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 unipotent matrices.

## Group properties

Property Satisfied Explanation
Abelian group No The matrices $\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}$ and $\begin{pmatrix} 1 & 0 \\ 1 & 1 \\\end{pmatrix}$ 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.