Special linear group of degree two

Definition

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:

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

The identity element is:

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

The inverse map is given by:

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

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.

Function	Value	Explanation
order	$\!q^{3}-q=q(q+1)(q-1)$	Kernel of determinant map from group of size $q(q+1)(q-1)^{2}$ surjecting to group of size $q-1$.
exponent	$p(q^{2}-1)$ if $p=2$, $\!p(q^{2}-1)/2$ if $p>2$	There are elements of order $p,q-1,q+1$, orders of all elements divide one of these.
number of conjugacy classes	$q+1$ if $p=2$, $q+4$ if $p>2$	For $p>2$, $q$ semisimple conjugacy classes (that do not split from $GL(2,q)$ and four conjugacy classes that merge into two in $GL(2,q)$.

Group properties

Property	Satisfied	Explanation
Abelian group	Yes if $q=2$, no otherwise
Nilpotent group	Yes if $q=2$, no otherwise	special linear group is perfect for $q\neq 2,3$, the case of $q=2,3$ can be checked.
Solvable group	Yes if $q=2,3$, no otherwise.	[special linear group is perfect]] for $q\neq 2,3$, the case of $q=2,3$ can be checked.
Supersolvable group	Yes if $q=2$, no otherwise	special linear group is perfect for $q\neq 2,3$, the case of $q=2,3$ can be checked.
Quasisimple group	Yes if $q\geq 4$	special linear group is quasisimple for $q\neq 2,3$.

Elements

Further information: Element structure of special linear group of degree two