Special linear group of degree two

Definition

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 $S{L}_2(R)$.

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:

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

The group operation is given by:

$\left(\begin{array}{cc}a& b\\ c& d\\ \end{array}\right)\left(\begin{array}{cc}a\text{'}& b\text{'}\\ c\text{'}& d\text{'}\\ \end{array}\right)=\left(\begin{array}{cc}aa\text{'}+bc\text{'}& ab\text{'}+bd\text{'}\\ ca\text{'}+dc\text{'}& cb\text{'}+dd\text{'}\\ \end{array}\right)$.

The identity element is:

$\left(\begin{array}{cc}1& 0\\ 0& 1\\ \end{array}\right)$.

The inverse map is given by:

${\left(\begin{array}{cc}a& b\\ c& d\\ \end{array}\right)}^{-1}=\left(\begin{array}{cc}d& -b\\ -c& a\\ \end{array}\right)$

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 $q$.

Arithmetic functions

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.

Function	Value	Similar groups	Explanation
order	$q^3-q=q(q+1)(q-1)$	The projective general linear group of degree two $PGL(2,q)$ has the same order, but is not isomorphic to it unless $q$ is a power of 2.	Kernel of determinant map from $GL(2,q)$, a group of size $q(q+1)(q-1{)}^2$ surjecting to ${\mathbb{F}}_q^{\ast }$, a group of size $q-1$. The order is thus $q(q+1)(q-1{)}^2/(q-1)=q(q+1)(q-1)$.
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 $q\ne 2,3$, the case of $q=2,3$ can be checked.
Solvable group	Yes if $q=2,3$, no otherwise.	special linear group is perfect for $q\ne 2,3$, the case of $q=2,3$ can be checked.
Supersolvable group	Yes if $q=2$, no otherwise	special linear group is perfect for $q\ne 2,3$, the case of $q=2,3$ can be checked.
Quasisimple group	Yes if $q\ge 4$	special linear group is quasisimple for $q\ne 2,3$.

Elements

Over a finite field

Further information: Element structure of special linear group of degree two over a finite field

Item	Value
order	$q^3-q=q(q-1)(q+1)$
exponent	$p(q^2-1)/2=p(q-1)(q+1)/2$ for $p$ odd, $2(q^2-1)$ for $p=2$
number of conjugacy classes	Case $q$ odd: $q+4$ Case $q$ even (and hence, a power of 2): $q+1$ equals the number of irreducible representations, see also linear representation theory of special linear group of degree two over a finite field
conjugacy class sizes	Case $q$ odd: 1 (2 times), $(q^2-1)/2$ (4 times), $q(q-1)$ ($(q-1)/2$ times), $q(q+1)$ ($(q-3)/2$ times) Case $q$ even: 1 (1 time), $q^2-1$ (1 time), $q(q-1)$ ($q/2$ times), $q(q+1)$ ($(q-2)/2$ times)
number of $p$-regular conjugacy classes (Where $p$ is the characteristic of the field)	$q$ equals the number of irreducible representations in that characteristic, see also modular representation theory of special linear group of degree two over a finite field in its defining characteristic
number of orbits under automorphism group	Case $q=p\ne 2$ (i.e., prime field for odd prime): $q+2$ (basically same as the conjugacy classes relative to $G{L}_2$) Case $q=p=2$: 3 Other cases: Complicated equals number of orbits of irreducible representations under automorphism group, see also linear representation theory of special linear group of degree two over a finite field

Linear representation theory

Over a finite field

Further information: Linear representation theory of special linear group of degree two over a finite field, modular representation theory of special linear group of degree two over a finite field in its defining characteristic

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

Item	Value
degrees of irreducible representations over a splitting field (such as $\overline{\mathbb{Q}}$ or $\mathbb{C}$)	Case $q$ odd: 1 (1 time), $(q-1)/2$ (2 times), $(q+1)/2$ (2 times), $q-1$ ($(q-1)/2$ times), $q$ (1 time), $q+1$ ($(q-3)/2$ times) Case $q$ even: 1 (1 time), $q-1$ ($q/2$ times), $q$ (1 time), $q+1$ ($(q-2)/2$ times)
number of irreducible representations	Case $q$ odd: $q+4$ Case $q$ even: $q+1$ See number of irreducible representations equals number of conjugacy classes, element structure of special linear group of degree two over a finite field#Conjugacy class structure
quasirandom degree (minimum degree of nontrivial irreducible representation)	Case $q$ odd: $(q-1)/2$ Case $q$ even: $q-1$
maximum degree of irreducible representation over a splitting field	$q+1$ if $q>3$ $q$ if $q\in \{2,3\}$
lcm of degrees of irreducible representations over a splitting field	Case $q=3$: We get 6 Case $q$ odd, $q>3$: $q(q+1)(q-1)/2=(q^3-q)/2$ Case $q$ even: $q(q+1)(q-1)=q^3-q$
sum of squares of degrees of irreducible representations over a splitting field	$q^3-q$, equal to the group order. See sum of squares of degrees of irreducible representations equals group 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 $p$):