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

This article gives specific information, namely, element structure, about a family of groups, namely: special linear group of degree two.
View element structure of group families | View other specific information about special linear group of degree two

This article describes the element structure of the special linear group of degree two over a finite field of order $q$ and characteristic $p$, where $q$ is a power of $p$. Some aspects of this discussion, with suitable infinitary analogues of cardinality, carry over to infinite fields and fields of infinite characteristic.

See also:

• Element structure of general linear group of degree two over a finite field
• Element structure of projective general linear group of degree two over a finite field
• Element structure of projective special linear group of degree two over a finite field

Summary

Item	Value
number of conjugacy classes	$q + 4$ if $q$ odd, $q + 1$ if $q$ a power of $2$ 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
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 $p = 2$

Particular cases

Group	$p$	$q$	order of the group	conjugacy class sizes (ascending order)	number of conjugacy classes (= $q + 4$ if $q$ odd, $q + 1$ if $q$ even)	element structure page
symmetric group:S3	2	2	6	1,2,3	3	element structure of symmetric group:S3
special linear group:SL(2,3)	3	3	24	1,1,4,4,4,4,6	7	element structure of special linear group:SL(2,3)
alternating group:A5	2	4	60	1,12,12,15,20	5	element structure of alternating group:A5
special linear group:SL(2,5)	5	5	120	1,1,12,12,12,12,20,20,30	9	element structure of special linear group:SL(2,5)
special linear group:SL(2,7)	7	7	336	1,1,24,24,24,24,42,42,42,56,56	11	element structure of special linear group:SL(2,7)
special linear group:SL(2,8)	2	8	504	1,56,56,56,56,63,72,72,72	9	element structure of special linear group:SL(2,8)
special linear group:SL(2,9)	3	9	720	1,1,40,40,40,40,72,72,72,72,90,90,90	13	element structure of special linear group:SL(2,9)

The expandable display contains the GAP code to verify this; the theoretical explanation is in the next section. [SHOW MORE]

All these functions use List and SL.

For the orders, we use Order:

gap> List([2,3,4,5,7,8,9],q -> [q,Order(SL(2,q))]);
[ [ 2, 6 ], [ 3, 24 ], [ 4, 60 ], [ 5, 120 ], [ 7, 336 ], [ 8, 504 ], [ 9, 720 ] ]

For the number of conjugacy classes, we use ConjugacyClasses and Length:

gap> List([2,3,4,5,7,8,9],q -> [q,Length(ConjugacyClasses(SL(2,q)))]);
[ [ 2, 3 ], [ 3, 7 ], [ 4, 5 ], [ 5, 9 ], [ 7, 11 ], [ 8, 9 ], [ 9, 13 ] ]

For the lists of sizes of conjugacy classes in ascending order, we use ConjugacyClasses, Size, and SortedList:

gap> List([2,3,4,5,7,8,9],q -> [q,SortedList(List(ConjugacyClasses(SL(2,q)),Size))]);
[ [ 2, [ 1, 2, 3 ] ], [ 3, [ 1, 1, 4, 4, 4, 4, 6 ] ], [ 4, [ 1, 12, 12, 15, 20 ] ], [ 5, [ 1, 1, 12, 12, 12, 12, 20, 20, 30 ] ],
  [ 7, [ 1, 1, 24, 24, 24, 24, 42, 42, 42, 56, 56 ] ], [ 8, [ 1, 56, 56, 56, 56, 63, 72, 72, 72 ] ],
  [ 9, [ 1, 1, 40, 40, 40, 40, 72, 72, 72, 72, 90, 90, 90 ] ] ]

Conjugacy class structure

When $p = 2$, there is a total of $q + 1$ conjugacy classes. When $p$ is odd, there is a total of $q + 4$ conjugacy classes.

In all cases, every element is either semisimple or unipotent or negative times unipotent (note that this statement is true only because of the small degree, two, and is not true for special linear groups of higher degree). There are $q$ conjugacy classes of semisimple elements, i.e., elements whose minimal polynomial has no repeated roots over its splitting field. The number of conjugacy classes of non-identity elements that are unipotent or negative times unipotent elements is $1$ when $p = 2$ and $4$ when $p$ is odd.

Note also that none of the semisimple conjugacy classes split in $SL_2$ relative to $GL_2$. The unipotent case is different for odd characteristic.

Summary for odd characteristic $p$, field size $q$

Nature of conjugacy class	Eigenvalues	Characteristic polynomial	Minimal polynomial	Size of conjugacy class	Number of such conjugacy classes	Total number of elements	Semisimple?	Diagonalizable over $\mathbb{F}_q$?	Splits in $SL_2$ relative to $GL_2$?
Diagonalizable over $\mathbb{F}_q$ with equal diagonal entries, hence a scalar	$\{ 1,1 \}$ or $\{ -1,-1\}$	$(x - a)^2$ where $a \in \{ -1,1 \}$	$x - a$ where $a \in \{ -1,1\}$	1	2	2	Yes	Yes	No
Not diagonal, has Jordan block of size two	$1$ (multiplicity 2) or $-1$ (multiplicity 2)	$(x - a)^2$ where $a \in \{ -1,1 \}$	Same as characteristic polynomial	$(q^2 - 1)/2$	4	$2(q^2 - 1)$	No	No	Yes (two conjugacy classes over $GL_2$, each splits into two over $SL_2$)
Diagonalizable over $\mathbb{F}_{q^2}$, not over $\mathbb{F}_q$. Must necessarily have no repeated eigenvalues.	Pair of conjugate elements of $\mathbb{F}_{q^2}$ of norm 1	$x^2 - ax + 1$, irreducible	Same as characteristic polynomial	$q(q - 1)$	$(q - 1)/2$	$q(q-1)^2/2 = (q^3 - 2q^2 + q)/2$	Yes	No	No
Diagonalizable over $\mathbb{F}_q$ with distinct (and hence mutually inverse) diagonal entries	$\lambda, 1/\lambda$ where $\lambda \in \mathbb{F}_q \setminus \{ 0,1,-1 \}$	$x^2 - (\lambda + 1/\lambda)x + 1$	Same as characteristic polynomial	$q(q+1)$	$(q - 3)/2$	$q(q+1)(q-3)/2 = (q^3 - 2q^2 - 3q)/2$	Yes	Yes	No
Total	NA	NA	NA	NA	$q + 4$	$q(q+1)(q-1) = q^3 - q$	$q(q^2 - 2q - 1) + 2$ elements $q$ conjugacy classes	$q(q^2 - 3q - 2)/2 + 2$ elements $(q + 1)/2$ conjugacy classes	$2(q^2 - 1)$elements 4 conjugacy classes

Summary for $p = 2$, $q$ a power of 2

Nature of conjugacy class	Eigenvalues	Characteristic polynomial	Minimal polynomial	Size of conjugacy class	Number of such conjugacy classes	Total number of elements	Semisimple?	Diagonalizable over $\mathbb{F}_q$?	Splits in $SL_2$ relative to $GL_2$?
Diagonalizable over $\mathbb{F}_q$ with equal diagonal entries, hence a scalar.	$1, 1$	$x^2 + 1$	$x + 1$	1	1	1	Yes	Yes	No
Diagonalizable over $\mathbb{F}_{q^2}$, not over $\mathbb{F}_q$. Must necessarily have no repeated eigenvalues.	Pair of conjugate elements of $\mathbb{F}_{q^2}$ of norm 1	$x^2 - ax + 1$, irreducible	$x^2 - ax + 1$, irreducible	$q(q - 1)$	$q/2$	$q^2(q - 1)/2 = (q^3 - q^2)/2$	Yes	No	No
Not diagonal, has Jordan block of size two	$1$ (multiplicity 2)	$x^2 + 1$	$x^2 + 1$	$q^2 - 1$	1	$q^2 - 1$	No	No	No
Diagonalizable over $\mathbb{F}_q$ with distinct (and hence mutually inverse) diagonal entries	$\lambda, 1/\lambda$ where $\lambda \in \mathbb{F}_q \setminus \{ 0,1 \}$	$x^2 - (\lambda + 1/\lambda)x + 1$	$x^2 - (\lambda + 1/\lambda)x + 1$	$q(q + 1)$	$(q - 2)/2$	$q(q+1)(q-2)/2 = (q^3 - q^2 - 2q)/2$	Yes	Yes	No
Total	NA	NA	NA	NA	$q + 1$	$q(q + 1)(q - 1) = q^3 - q$	$(q + 1)(q - 1)^2 = q^3 - q^2 - q + 1$

Automorphism class structure

We have that special linear group of degree two has a class-inverting automorphism. In particular, any such group is a group in which every element is automorphic to its inverse. Also, special linear group of degree two is ambivalent iff -1 is a square, where an ambivalent group is a group in which every element is conjugate to its inverse.

We discuss below the automorphism class structure, i.e., the orbit structure under the action of the automorphism group.

Summary for odd characteristic $p$, field size $q$

We let $q = p^r$. Note that if $q = p$, then $r = 1$.

Nature of automorphism class	Number of automorphism classes of this type	Number of conjugacy classes within each automorphism class	Size of each conjugacy class	Size of each automorphism class	Total number of elements across all such automorphism classes
Diagonalizable over $\mathbb{F}_q$ with equal diagonal entries, hence a scalar	2	1	1	1	2
Not diagonal, has Jordan block of size two	2	2	$(q^2 - 1)/2$	$q^2 - 1$	$2(q^2 - 1)$
Diagonalizable over $\mathbb{F}_{q^2}$, not over $\mathbb{F}_q$. Must necessarily have no repeated eigenvalues.	(complicated, depends on $r$)	(complicated, depends on $r$)	$q(q - 1)$		$q(q-1)^2/2 = (q^3 - 2q^2 + q)/2$
Diagonalizable over $\mathbb{F}_q$ with distinct (and hence mutually inverse) diagonal entries	(complicated, depends on $r$)	(complicated, depends on $r$)	$q(q + 1)$		$q(q+1)(q-3)/2 = (q^3 - 2q^2 - 3q)/2$

Central elements

The center is a subgroup of order either 1 or 2, depending on whether $q$ is even or odd. For odd $q$, the center is given by:

$\left\{ \begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}, \begin{pmatrix} -1 & 0 \\ 0 & -1 \\\end{pmatrix} \right\}$

For even $q$ (i.e., the characteristic is 2 and $q$ is a power of 2), the center is the trivial group, comprising only the identity element.

Jordan block of size two

The conjugacy classes of this type are the only ones that split in the special linear group relative to the general linear group (in the odd characteristic case), i.e., where there are elements of $SL_2$ that are conjugate in $GL_2$ but not in $SL_2$.

Over $GL_2$, there are two conjugacy classes in odd characteristic (which collapse to one class in even characteristic):

• The conjugacy class of $\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}$. In particular, this conjugacy class includes all matrices of the form $\begin{pmatrix} 1 & \lambda \\ 0 & 1 \\\end{pmatrix}$ and also all matrices of the form $\begin{pmatrix} 1 & 0 \\ \lambda & 1 \\\end{pmatrix}$ where $\lambda \in \mathbb{F}_q^\ast$.
• The conjugacy class of $\begin{pmatrix} -1 & 1 \\ 0 & -1 \\\end{pmatrix}$. In particular, this conjugacy class includes all matrices of the form $\begin{pmatrix} -1 & \lambda \\ 0 & -1 \\\end{pmatrix}$ and also all matrices of the form $\begin{pmatrix} -1 & 0 \\ \lambda & -1 \\\end{pmatrix}$ where $\lambda \in \mathbb{F}_q^\ast$.

Characteristic 2 case

In this case, both conjugacy classes collapse into a single conjugacy class. We have:

Item	Value	Explanation
Trace of conjugacy class	0
Norm or determinant of conjugacy class	1
Minimal polynomial of conjugacy class	$(x - 1)^2$	Repeated eigenvalue
Characteristic polynomial of conjugacy class	$(x - 1)^2 = x^2 + 1$
Centralizer of any element in conjugacy class	Isomorphic to additive group of $\mathbb{F}_q$	For the element $\begin{pmatrix}1 & 1 \\ 0 & 1 \\\end{pmatrix}$, the centralizer is all matrices of the form $\begin{pmatrix} 1 & t \\ 0 & 1 \\\end{pmatrix}$ where $t \in \mathbb{F}_q$.
Size of conjugacy class	$q(q - 1)(q + 1)/q = (q - 1)(q + 1) = q^2 - 1$	size of conjugacy class equals index of centralizer

Here is the information on the collection of all conjugacy classes:

Item	Value	Explanation
number of conjugacy classes of this type	1
total number of elements of this type	$q^2 - 1$

Odd characteristic case

In this case, each conjugacy class splits further into two, giving a total of four conjugacy classes. We have:

Item	Value	Explanation
Trace of conjugacy class	2 or -2, depending on whether the eigenvalues are 1 or -1
Norm or determinant of conjugacy class	1
Minimal polynomial of conjugacy class	$x - 1$ or $x + 1$, depending on whether the eigenvalues are 1 or -1	Repeated eigenvalue
Characteristic polynomial of conjugacy class	$(x - 1)^2 = x^2 - 2x + 1$ or $(x + 1)^2 = x^2 + 2x + 1$
Centralizer of any element in conjugacy class	Isomorphic to direct product of additive group of $\mathbb{F}_q$ and cyclic group of order two	For the element $\begin{pmatrix}1 & 1 \\ 0 & 1 \\\end{pmatrix}$, the centralizer is all matrices of the form $\begin{pmatrix} 1 & t \\ 0 & 1 \\\end{pmatrix}$ and all matrices of the form $\begin{pmatrix} -1 & t \\ 0 & -1 \\\end{pmatrix}$ where $t \in \mathbb{F}_q$
Size of conjugacy class	$q(q - 1)(q + 1)/(2q) = (q - 1)(q + 1)/2 = (q^2 - 1)/2$	size of conjugacy class equals index of centralizer

Given two matrices:

$\begin{pmatrix} 1 & a \\ 0 & 1 \\\end{pmatrix}, \begin{pmatrix} 1 & b \\ 0 & 1 \\\end{pmatrix}, \qquad a,b \in \mathbb{F}_q^\ast$

the following rule works to determine whether they are conjugate in the special linear group: the elements are conjugate if and only if $a/b$ is a square in $\mathbb{F}_q^\ast$.

Here is the information on the collection of all conjugacy classes:

Item	Value	Explanation
number of conjugacy classes of this type	4	each of the two conjugacy classes relative to $GL_2$ splits into two in $SL_2$.
total number of elements of this type	$2(q^2 - 1)$

Elements diagonalizable over $\mathbb{F}_{q^2}$ but not $\mathbb{F}_q$

These elements have pairs of distinct eigenvalues over $\mathbb{F}_{q^2}$ that are conjugate over $\mathbb{F}_q$. The unique non-identity automorphism of $\mathbb{F}_{q^2}$ over $\mathbb{F}_q$ is the map $x \mapsto x^q$, so these two elements are $q^{th}$ powers of each other, i.e., if one of them is $\alpha$, the other one is $\alpha^q$.

The conjugacy class is parameterized by the pair $\{ \alpha, \alpha^q \}$.

The element must have determinant 1, so this forces $(\alpha)(\alpha^q) = 1$, so $\alpha^{q+1} = 1$ and $\alpha^q = \alpha^{-1}$.

Here is more information:

Item	Value	Explanation
Trace of conjugacy class	$a = \alpha + \alpha^q = \alpha + \alpha^{-1}$, is an element of $\mathbb{F}_q$	Trace is sum of eigenvalues
Norm or determinant of conjugacy class	1
Minimal polynomial of conjugacy class	$x^2 - ax + 1$
Characteristic polynomial of conjugacy class	Same as minimal polynomial
Centralizer of any element of conjugacy class	It is the intersection of the multiplicative group $\mathbb{F}_{q^2}^\ast$ which is a cyclic group of order $q^2 - 1$ (acting as linear transformations by viewing $\mathbb{F}_{q^2}$ as a two-dimensional vector space over $\mathbb{F}_q$), and the special linear group. This becomes a cyclic subgroup of order $q + 1$ in $\mathbb{F}_{q^2}^\ast$.
Size of conjugacy class	$q(q+1(q-1)/(q+1) = q(q-1)$	Equals the index of the centralizer

Here is information on the collection of all such conjugacy classes:

Item	Value	Explanation
number of conjugacy classes of this type	$(q - 1)/2$ if $q$ odd, $q/2$ if $q$ even	First explanation: The conjugacy classes are classified by pairs $\{ \alpha, \alpha^q \}$ where $\alpha^{q + 1} = 1$ and $\alpha \notin \mathbb{F}_q$. There are $q + 1$ elements of order dividing $q + 1$ in $\mathbb{F}_q^\ast$ (because it is cyclic). Of these, depending on whether $q$ is even or odd, 1 or 2 elements live inside $\mathbb{F}_q^\ast$. The remaining elements occur in pairs $\{\alpha,\alpha^q \}$, so we get $(q + 1 - (1 \operatorname{or} 2))/2$ which simplifies based on the cases. Second explanation: There are $q$ monic polynomials of degree 2 with constant term 1. We need to figure out how many of them are irreducible, and we can do this by subtracting the number that already factor. When $q$ is odd, we get $2 + (q - 3)/2 = (q + 1)/2$ polynomials that factor over $\mathbb{F}_q$, so the difference is $q - (q +1)/2 = (q - 1)/2$. When $q$ is even, we get $1 + (q - 2)/2 = q/2$ polynomials that factor, so the difference is $q - q/2 = q/2$. (Addendum to second explanation): The $2 + (q - 3)/2$ and $1 + (q - 2)/2$ arise as (number of elements that square to 1) + (number of distinct pairs of element and its inverse)
total number of elements of this type	$q(q-1)^2/2 = (q^3 - 2q^2 + q)/2$ if $q$ odd, $q^2(q - 1)/2 = (q^3 - q^2)/2$ if $q$ even	Multiply the size of each conjugacy class with the number of conjugacy classes of this type

Elements diagonalizable over $\mathbb{F}_q$ with distinct and hence mutually inverse entries

Each such conjugacy class is specified by an unordered pair of distinct elements of $\mathbb{F}_q^\ast$, say $\lambda, \lambda^{-1}$. Note that the eigenvalues must be inverses because their product needs to be 1 for the element to be in the special linear group.

Item	Value	Explanation
Trace of conjugacy class	$a = \lambda + \lambda^{-1}$, is an element of $\mathbb{F}_q$	Trace is sum of eigenvalues
Norm or determinant of conjugacy class	1
Minimal polynomial of conjugacy class	$(x - \lambda)(x - \lambda^{-1})$ which equals $x^2 - ax + 1$, $a$ the trace as above.	Distinct eigenvalues
Characteristic polynomial of conjugacy class	Same as minimal polynomial	Distinct eigenvalues
Centralizer of any element of conjugacy class	It is isomorphic to $\mathbb{F}_q^\ast$, which is a group of order $(q - 1)$	This is easiest to see for the representative that is diagonal: its centralizer is the subgroup of diagonal matrices of determinant 1, i.e., matrices of the form $\begin{pmatrix} s & 0 \\ 0 & s^{-1} \\\end{pmatrix}$.
Size of conjugacy class	$(q - 1)q(q + 1)/(q - 1) = q(q + 1)$	It equals the index of the centralizer

Here is combined information for all conjugacy classes:

Item	Value	Explanation
number of conjugacy classes of this type	$(q - 3)/2$ for odd $q$, $(q - 2)/2$ for $q$ a power of 2	We consider $\mathbb{F}_q^\ast$ (order $q - 1$) and remove the elements that are equal to their own inverse (1 such element in characteristic 2, 2 such elements in characteristic 3). Then, we divide by 2 because we are interested in unordered pairs only.
total number of elements of this type	$q(q+1)(q-3)/2= (q^3 - 2q^2 - 3q)/2$ if $q$ odd, $q(q+1)(q-2)/2 = (q^3 - q^2 - 2q)/2$ if $q$ even	Multiply number of conjugacy classes and size of each conjugacy class.