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

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

Summary

Item	Value
conjugacy class sizes	Case $q$ congruent to 1 mod 4 (e.g., $q = 5,9,13,17,25,29$): 1 (1 time), $(q^2 - 1)/2$ (2 times), $q(q+1)/2$ (1 time), $q(q-1)$ ($(q - 1)/4$ times), $q(q+1)$ ($(q - 5)/4$ times) Case $q$ congruent to 3 mod 4 (e.g., $q = 3,7,11,19,23,27,31$: 1 (1 time), $q(q-1)/2$ (1 time), $(q^2 - 1)/2$ (2 times), $q(q - 1)$ ($(q - 3)/4$ times), $q(q + 1)$ ($(q - 3)/4$ times Case $q$ even (e.g., $q = 2,4,8,16,32$): 1 (1 time), $q(q - 1)$ ($q/2$ times), $q^2 - 1$ (1 time), $q(q+1)$ ($(q-2)/2$ times)
number of conjugacy classes	Case $q$ odd :$(q + 5)/2$ Case $q$ even: $q + 1$ equals number of irreducible representations, see also linear representation theory of projective special linear group of degree two over a finite field
number of $p$-regular conjugacy classes (where $p$ is the characteristic of the field)	$(q + 1)/2$ equals the number of irreducible representations in that characteristic, see also modular representation theory of projective special linear group of degree two over a finite field in its defining characteristic
order	General formula: $(q^3 - q)/\operatorname{gcd}(2,q-1)$ Case $q$ odd: $(q^3 - q)/2 = (q-1)q(q+1)/2$ Case $q$ even: $q^3 - q$
exponent	Case $q$ odd: $p(q^2 - 1)/4$ Case $q$ even: $2(q^2 - 1)$

Particular cases

$q$ (field size)	$p$ (underlying prime, field characteristic)	Case on $q$	Group	order of the group (= $(q^3 - q)/2$ if $q$ odd, $q^3 - q$ if $q$ even)	sizes of conjugacy classes (ascending order)	number of conjugacy classes (= $(q + 5)/2$ if $q$ odd, $q + 1$ if $q$ even)	element structure page
2	2	even	symmetric group:S3	6	1,2,3	3	element structure of symmetric group:S3
3	3	3 mod 4	alternating group:A4	12	1,3,4,4	4	element structure of alternating group:A4
4	2	even	alternating group:A5	60	1,12,15,20,20	5	element structure of alternating group:A5
5	5	1 mod 4	alternating group:A5	60	1,12,15,20,20	5	element structure of alternating group:A5
7	7	3 mod 4	projective special linear group:PSL(3,2)	168	1,21,24,24,42,56	6	element structure of projective special linear group:PSL(3,2)
8	2	even	projective special linear group:PSL(2,8)	504	1,56,56,56,56,63,72,72,72	9	element structure of projective special linear group:SL(2,8)
9	3	1 mod 4	alternating group:A6	360	1,40,40,45,72,72,90	7	element structure of alternating group:A6

Conjugacy class structure

See also element structure of special linear group of degree two#Conjugacy class structure.

Number of conjugacy classes

As we know in general, number of conjugacy classes in projective 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).

Value of $\operatorname{gcd}(2,q-1)$	Corresponding congruence classes of $q$ mod 2	Number of conjugacy classes (polynomial of degree 2 - 1 = 1 in $q$)	Additional comments
1	0 mod 2 (e.g., $q = 2,4,8,\dots$)	$q + 1$	In this case, we have an isomorphism between linear groups when degree power map is bijective, so $SL(2,q) \cong PGL(2,q) \cong PSL(2,q)$
2	1 mod 2 (e.g., $q = 3,5,7,\dots$)	$(q + 5)/2$

Case where $p$ is odd, $4$ divides $q - 1$

Nature of conjugacy class upstairs in $SL_2$	Eigenvalues	Characteristic polynomial	Minimal polynomial	Size of conjugacy class	Number of such conjugacy classes	Total number of elements	Orders of elements
Diagonalizable over $\mathbb{F}_q$ with equal diagonal entries, hence a scalar	$\{ 1,1 \}$ or $\{ -1,-1\}$, both correspond to the same element	$(x - a)^2$ where $a \in \{ -1,1 \}$	$x - a$ where $a \in \{ -1,1\}$	1	1	1	1
Not diagonal, has Jordan block of size two	$1$ (multiplicity 2) or $-1$ (multiplicity 2). Each conjugacy class has one representative of each type.	$(x - a)^2$ where $a \in \{ -1,1 \}$	Same as characteristic polynomial	$(q^2 - 1)/2$	2	$q^2 - 1$	$p$
Diagonalizable over $\mathbb{F}_q$ with diagonal entries squaring to $-1$	$i, -i$ where $i^2 = -1$	$x^2 + 1$	$x^2 + 1$	$q(q + 1)/2$	1	$q(q+1)/2$	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. Each pair identified with its negative pair.	$x^2 - ax + 1$, irreducible; note that $x^2 - ax + 1$'s pair and $x^2 + ax +1$'s pair get identified.	Same as characteristic polynomial	$q(q - 1)$	$(q - 1)/4$	$q(q - 1)^2/4 = (q^3 - 2q^2 + q)/4$	divisor of $q + 1$, maximum at $q + 1$
Diagonalizable over $\mathbb{F}_q$ with distinct (and hence mutually inverse) diagonal entries, whose square is not $-1$	$\lambda, 1/\lambda$ where $\lambda \in \mathbb{F}_q \setminus \{ 0,1,-1,i,-i \}$ where $i,-i$ are square roots of $-1$. Note that the representative pairs $\{ \lambda, 1/\lambda \}$ and $\{ -\lambda,-1/\lambda \}$ get identified.	$x^2 - (\lambda + 1/\lambda)x + 1$, again with identification.	$x^2 - (\lambda + 1/\lambda)x + 1$, again with identification.	$q(q + 1)$	$(q - 5)/4$	$q(q+1)(q-5)/4 = (q^3 - 4q^2 - 5q)/4$	divisor of $(q - 1)/2$, maximum at $(q - 1)/2$ for $q > 5$
Total	NA	NA	NA	NA	$(q + 5)/2$	$(q^3 - q)/2$

Case where $p$ is odd, $4$ does not divides $q - 1$

Nature of conjugacy class upstairs in $SL_2$	Eigenvalues	Characteristic polynomial	Minimal polynomial	Size of conjugacy class	Number of such conjugacy classes	Total number of elements	Order of elements
Diagonalizable over $\mathbb{F}_q$ with equal diagonal entries, hence a scalar	$\{ 1,1 \}$ or $\{ -1,-1\}$, both correspond to the same element	$(x - a)^2$ where $a \in \{ -1,1 \}$	$x - a$ where $a \in \{ -1,1\}$	1	1	1	1
Diagonalizable over $\mathbb{F}_{q^2}$, not over $\mathbb{F}_q$, eigenvalues square roots of $-1$	Square roots of $-1$	$x^2 + 1$	$x^2 + 1$	$q(q - 1)/2$	1	$q(q-1)/2 = (q^2 - q)/2$	2
Not diagonal, has Jordan block of size two	$1$ (multiplicity 2) or $-1$ (multiplicity 2). Each conjugacy class has one representative of each type.	$(x - a)^2$ where $a \in \{ -1,1 \}$	Same as characteristic polynomial	$(q^2 - 1)/2$	2	$q^2 - 1$	$p$
Diagonalizable over $\mathbb{F}_{q^2}$, not over $\mathbb{F}_q$. Must necessarily have no repeated eigenvalues, eigenvalues not square roots of $-1$.	Pair of conjugate elements of $\mathbb{F}_{q^2}$ of norm 1, not square roots of -1. Each pair identified with its negative pair.	$x^2 - ax + 1$, $a \ne 0$ irreducible; note that $x^2 - ax + 1$'s pair and $x^2 + ax +1$'s pair get identified.	Same as characteristic polynomial	$q(q - 1)$	$(q - 3)/4$	$q(q - 1)(q - 3)/4 = (q^3 - 4q^2 + 3q)/4$	divisor of $q + 1$, maximum of $q + 1$ occurs.
Diagonalizable over $\mathbb{F}_q$ with distinct (and hence mutually inverse) diagonal entries	$\lambda, 1/\lambda$ where $\lambda \in \mathbb{F}_q \setminus \{ 0,1,-1 \}$. Note that the representative pairs $\{ \lambda, 1/\lambda \}$ and $\{ -\lambda,-1/\lambda \}$ get identified. However, the pair corresponding to the two square roots of $-1$ equals its own negative.	$x^2 - (\lambda + 1/\lambda)x + 1$, again with identification.	$x^2 - (\lambda + 1/\lambda)x + 1$, again with identification.	$q(q + 1)$	$(q - 3)/4$	$q(q+1)(q-3)/4 = (q^3 - 2q^2 - 3q)/4$	divisor of $(q - 1)/2$, maximum of $(q - 1)/2$ occurs
Total	NA	NA	NA	NA	$(q + 5)/2$	$(q^3 - q)/2$

Case where $p = 2$

In this case, the natural surjective map from the special linear group of degree two to the projective special linear group of degree two is an isomorphism, so the conjugacy class structure of both groups is the same. Details below:

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	$q + 1$	$q(q + 1)(q - 1) = q^3 - q$	$(q + 1)(q - 1)^2 = q^3 - q^2 - q + 1$