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 $SL_2(R)$.

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:

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

The identity element is:

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

The inverse map is given by:

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

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 $q$.

Note that for a finite field, we have the following: special unitary group of degree two equals special linear group of degree two over a finite field. In other words, $SL(2,q)$ is isomorphic to $SU(2,q)$.

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 $q(q+1)(q-1)^2/(q - 1) = q(q+1)(q-1)$.
See order formulas for linear groups of degree two for more information.
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

As before, $q$ is the field size and $p$ is the characteristic of the field, so $p$ is a prime number and $q$ is a power of $p$.

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 $GL_2$)
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

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 $p$):