Special linear group of degree two

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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

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):