Element structure of special linear group:SL(2,3)

This article gives specific information, namely, element structure, about a particular group, namely: special linear group:SL(2,3).
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This article gives detailed information about the element structure of special linear group:SL(2,3).

See also element structure of special linear group of degree two over a finite field.

Conjugacy and automorphism class structure

Conjugacy classes

Note that since we are over field:F3, $-1=2$, so all the $-1$s below can be rewritten as $2$s.

Conjugacy class representative	Conjugacy class size	List of all elements of conjugacy class	Order of elements in conjugacy class
$\left(\begin{array}{cc}1& 0\\ 0& 1\\ \end{array}\right)$	1	$\left(\begin{array}{cc}1& 0\\ 0& 1\\ \end{array}\right)$	1
$\left(\begin{array}{cc}-1& 0\\ 0& -1\\ \end{array}\right)$	1	$\left(\begin{array}{cc}-1& 0\\ 0& -1\\ \end{array}\right)$	2
$\left(\begin{array}{cc}1& 1\\ 0& 1\\ \end{array}\right)$	4	[SHOW MORE] $\left(\begin{array}{cc}1& 1\\ 0& 1\\ \end{array}\right)$, $\left(\begin{array}{cc}-1& 1\\ -1& 0\\ \end{array}\right)$, $\left(\begin{array}{cc}1& 0\\ -1& 1\\ \end{array}\right)$, $\left(\begin{array}{cc}0& 1\\ -1& -1\\ \end{array}\right)$	3
$\left(\begin{array}{cc}1& -1\\ 0& 1\\ \end{array}\right)$	4	[SHOW MORE] $\left(\begin{array}{cc}1& -1\\ 0& 1\\ \end{array}\right)$, $\left(\begin{array}{cc}-1& -1\\ 1& 0\\ \end{array}\right)$, $\left(\begin{array}{cc}1& 0\\ 1& 1\\ \end{array}\right)$, $\left(\begin{array}{cc}0& -1\\ 1& -1\\ \end{array}\right)$	3
$\left(\begin{array}{cc}-1& 1\\ 0& -1\\ \end{array}\right)$	4	[SHOW MORE] $\left(\begin{array}{cc}-1& 1\\ 0& -1\\ \end{array}\right)$, $\left(\begin{array}{cc}1& 1\\ -1& 0\\ \end{array}\right)$, $\left(\begin{array}{cc}-1& 0\\ -1& -1\\ \end{array}\right)$, $\left(\begin{array}{cc}0& 1\\ -1& 1\\ \end{array}\right)$	6
$\left(\begin{array}{cc}-1& -1\\ 0& -1\\ \end{array}\right)$	4	[SHOW MORE] $\left(\begin{array}{cc}-1& -1\\ 0& -1\\ \end{array}\right)$, $\left(\begin{array}{cc}1& -1\\ 1& 0\\ \end{array}\right)$, $\left(\begin{array}{cc}-1& 0\\ 1& -1\\ \end{array}\right)$, $\left(\begin{array}{cc}0& -1\\ 1& 1\\ \end{array}\right)$	6
$\left(\begin{array}{cc}0& -1\\ 1& 0\\ \end{array}\right)$	6	[SHOW MORE] $\left(\begin{array}{cc}0& -1\\ 1& 0\\ \end{array}\right)$, $\left(\begin{array}{cc}0& 1\\ -1& 0\\ \end{array}\right)$, $\left(\begin{array}{cc}1& 1\\ 1& -1\\ \end{array}\right)$, $\left(\begin{array}{cc}-1& 1\\ 1& 1\\ \end{array}\right)$, $\left(\begin{array}{cc}1& -1\\ -1& -1\\ \end{array}\right)$, $\left(\begin{array}{cc}-1& -1\\ -1& 1\\ \end{array}\right)$	4

Automorphism classes

Below are the orbits under the action of the automorphism group, i.e., the automorphism classes of elements of the group.

List of representatives for each conjugacy class in the automorphism class	Number of conjugacy classes in the automorphism class	Size of each conjugacy class	Automorphism class size	Order of elements in conjugacy class
$\left(\begin{array}{cc}1& 0\\ 0& 1\\ \end{array}\right)$	1	1	1	1
$\left(\begin{array}{cc}-1& 0\\ 0& -1\\ \end{array}\right)$	1	1	1	2
$\left(\begin{array}{cc}1& 1\\ 0& 1\\ \end{array}\right)$, $\left(\begin{array}{cc}1& -1\\ 0& 1\\ \end{array}\right)$	2	4	8	3
$\left(\begin{array}{cc}-1& 1\\ 0& -1\\ \end{array}\right)$, $\left(\begin{array}{cc}-1& -1\\ 0& -1\\ \end{array}\right)$	2	4	8	6
$\left(\begin{array}{cc}0& -1\\ 1& 0\\ \end{array}\right)$	1	6	6	4

Relationship with conjugacy class structure for an arbitrary special linear group of degree two

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

Nature of conjugacy class	Eigenvalue pairs of all conjugacy classes	Characteristic polynomials of all conjugacy classes	Minimal polynomials of all conjugacy classes	Size of conjugacy class (generic $q$)	Size of conjugacy class ($q=3$)	Number of such conjugacy classes (generic $q$)	Number of such conjugacy classes ($q=3$)	Total number of elements (generic $q$)	Total number of elements ($q=3$)	Representative matrices (one per conjugacy class)
Scalar	$\{1,1\}$ or $\{-1,-1\}$	$x^2-2x+1$ or $x^2-x+1$	$x-1$ or $x+1$	1	1	2	2	2	2	$\left(\begin{array}{cc}1& 0\\ 0& 1\\ \end{array}\right)$ and $\left(\begin{array}{cc}-1& 0\\ 0& -1\\ \end{array}\right)$
Not diagonal, Jordan block of size two	$\{1,1\}$ or $\{-1,-1\}$	$x^2-2x+1$ or $x^2-x+1$	$x^2-2x+1$ or $x^2-x+1$	$(q^2-1)/2$	4	4	4	$2(q^2-1)$	16	[SHOW MORE] $\left(\begin{array}{cc}1& 1\\ 0& 1\\ \end{array}\right)$, $\left(\begin{array}{cc}1& -1\\ 0& 1\\ \end{array}\right)$, $\left(\begin{array}{cc}-1& 1\\ 0& -1\\ \end{array}\right)$, $\left(\begin{array}{cc}-1& -1\\ 0& -1\\ \end{array}\right)$
Diagonalizable over field:F9, not over field:F3. Must necessarily have no repeated eigenvalues.	pair of square roots of $-1$ in field:F9	$x^2+1$	$x^2+1$	$q(q-1)$	6	$(q-1)/2$	1	$q(q-1{)}^2/2$	6	$\left(\begin{array}{cc}0& -1\\ 1& 0\\ \end{array}\right)$
Diagonalizable over field:F3 with distinct diagonal entries	--	--	--	$q(q+1)$	12	$(q-3)/2$	0	$q(q+1)(q-3)/2$	0	--
Total	NA	NA	NA	NA	NA	$q+4$	7	$q^3-q$	24	NA