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

This article describes the element structure of the specific information about::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.

Generalizing to infinite fields and non-fields

 * Element structure of special linear group of degree two over a field
 * Element structure of special linear group of degree two over a finite discrete valuation ring

Related groups over finite fields

 * 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

Particular cases
The expandable display contains the GAP code to verify this; the theoretical explanation is in the next section.

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 ] ] ]

Number of conjugacy classes
As we know in general, number of conjugacy classes in special linear group of fixed degree over a finite field is PORC function of field size, the degree of this PORC function is one less than the degree of matrices, and we make cases based on the congruence classes modulo the degree of matrices. Thus, we expect that the number of conjugacy classes is a PORC function of the field size of degree 2 - 1 = 1, and we need to make cases based on the congruence class of the field size modulo 2. Moreover, the general theory also tells us that the polynomial function of $$q$$ depends only on the value of $$\operatorname{gcd}(n,q-1)$$, which in turn can be determined by the congruence class of $$q$$ mod $$n$$ (with $$n = 2$$ here).

General strategy and summary
Before making the entire table, we recall the general strategy: first, imitate the procedure of element structure of general linear group of degree two over a finite field to determine that $$GL(2,q)$$-conjugacy classes in $$SL(2,q)$$. Then, use the splitting criterion for conjugacy classes in the special linear group to determine which of these conjugacy classes split, and how much.

In this case, the fact that conjugacy class of elements with semisimple generalized Jordan block does not split in special linear group over a finite field tells us that the only conjugacy class that might split is the conjugacy class with a Jordan block of size two. Further, the splitting criterion for conjugacy classes in special linear group of prime degree over a finite field, applied to the case of degree two, tells us that:


 * When the field size is odd, there are two such $$GL(2,q)$$-conjugacy classes (repeated eigenvalue 1, and repeated eigenvalue -1) and each splits into two $$SL(2,q)$$-conjugacy classes. We thus get a total of 4 $$SL(2,q)$$-conjugacy classes of this type.
 * When the field size is even, there is only one such $$GL(2,q)$$-conjugacy class and it does not split over $$SL(2,q)$$.

Summary for odd field size
The key feature for fields of odd size is that there exist two distinct square roots of unity in such fields: 1 and -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 $$r = 1$$.

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:

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

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

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:

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:

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

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.

Here is combined information for all conjugacy classes: