Projective special linear group of degree two

Definition

For a field or commutative unital ring

The projective special linear group of degree two over a field $k$, or more generally over a commutative unital ring $R$, is defined as the quotient of the special linear group of degree two over the same field or commutative unital ring by the subgroup of scalar matrices in that group. The group is denoted by $PSL(2,R)$ or $PSL_2(R)$.

For a prime power

Suppose $q$ is a prime power. The projective special linear group $PSL(2,q)$ is defined as the projective special linear group of degree two over the field (unique up to isomorphism) with $q$ elements.

Particular cases

For prime powers $q$

Note that for $q = 2$, $PSL(2,q) \cong PGL(2,q) \cong SL(2,q) \cong GL(2,q)$. Also, for $q$ a power of 2 (so $p = 2$), $PSL(2,q) \cong PGL(2,q) \cong SL(2,q)$ and $GL(2,q) \cong PSL(2,q) \times \mathbb{F}_q^\ast$.

Field size $q$	Field characteristic $p$	Exponent on $p$ giving $q$	Group $PSL(2,q)$	Order: $q(q+1)(q-1)/\operatorname{gcd}(2,q-1)$	Second part of GAP ID (if applicable)	Proof of isomorphism	Comments
2	2	1	symmetric group:S3	6	1	PGL(2,2) is isomorphic to S3	not simple (one of two exceptions)
3	3	1	alternating group:A4	12	3	PSL(2,3) is isomorphic to A4	not simple (one of two exceptions)
4	2	2	alternating group:A5	60	5	PGL(2,4) is isomorphic to A5	minimal simple group
5	5	1	alternating group:A5	60	5	PSL(2,5) is isomorphic to A5	minimal simple group
7	7	1	projective special linear group:PSL(3,2)	168	42	PSL(2,7) is isomorphic to PSL(3,2)	minimal simple group
8	2	3	projective special linear group:PSL(2,8)	504	156	note that it is also the same as $PGL(2,8)$ and $SL(2,8)$	minimal simple group
9	3	2	alternating group:A6	360	114	PSL(2,9) is isomorphic to A6	simple non-abelian group but not a minimal simple group; contains alternating group:A5. See classification of finite minimal simple groups.
11	11	1	projective special linear group:PSL(2,11)	660	13	--	simple non-abelian group but not a minimal simple group -- contains alternating group:A5
13	13	1	projective special linear group:PSL(2,13)	1092	25	--	minimal simple group
16	2	4	projective special linear group:PSL(2,16)	4080	--	--	simple non-abelian group but not a minimal simple group -- contains alternating group:A5 as the subgroup $PSL(2,4)$
17	17	1	projective special linear group:PSL(2,17)	2448	--	--	minimal simple group

Arithmetic functions

Below we give the arithmetic functions for $PSL(2,q)$, where $q$ is a power $p^r$ of a prime $p$.

Function	Value	Explanation
order	$\frac{q^3 - q}{\operatorname{gcd}(2,q-1)} = \frac{q(q-1)(q+1)}{\operatorname{gcd}(2,q-1)}$ Becomes $(q^3 - q)/2$ for odd $q$, $q^3 - q$ when $p = 2$	See order formulas for linear groups of degree two. Also, see element structure of projective special linear group of degree two
number of conjugacy classes	$(q + 5)/2$ for odd $q$, $q + 1$ when $p = 2$	See element structure of projective special linear group of degree two

Group properties

The property listings below are for $PSL(2,q)$, $q$ a prime power.

Property	Satisfied?	Explanation
abelian group	No (never)
nilpotent group	No (never)
solvable group	No (never)
simple group, simple non-abelian group	Yes (almost always)	Exceptions: $q = 2$ (we get symmetric group:S3) and $q = 3$ (we get alternating group:A4). See projective special linear group is simple.
minimal simple group	Sometimes	See classification of finite minimal simple groups. Minimal simple in precisely these cases: $q = 2^r$, $r$ prime; $q = 3^r$, $r$ an odd prime; $q = p$ is a prime greater than 3 such that $5$ divides $p^2 + 1$, and $q = 5$ (which is not necessary to add, since $PSL(2,5) \cong PSL(2,4)$ so it gets double-counted). In particular, the following values of $q$ give simple non-abelian groups that are not minimal simple: $q = 9$ (gives alternating group:A6), $11, 16, 19, \dots$.

Elements

Over a finite field

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

Below is a 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: $2(q^2 - 1)$

Linear representation theory

Over a finite field

Further information: linear representation theory of projective special linear group of degree two over a finite field

Below is a summary:

Item	Value
degrees of irreducible representations over a splitting field (such as $\overline{\mathbb{Q}}$ or $\mathbb{C}$)	Case $q$ congruent to 1 mod 4 (e.g., $q=5,9,13,17,25,29$): 1 (1 time), $(q + 1)/2$ (2 times), $q - 1$ ($(q - 1)/4$ times), $q$ (1 time), $q + 1$ ($(q - 5)/4$ times) Case $q$ congruent to 3 mod 4 (e.g., $q = 3,7,11,19,23,27$): 1 (1 time), $(q - 1)/2$ (2 times), $q - 1$ ($(q - 3)/4$ times), $q$ (1 time), $q + 1$ ($(q - 3)/4$ times) Case $q$ even (e.g., $q=2,4,8,16,32$): 1 (1 time), $q - 1$ ($q/2$ times), $q$ (1 time), $q + 1$ ($(q - 2)/2$ times)
number of irreducible representations	Case $q$ odd: $(q + 5)/2$; Case $q$ even: $q + 1$. See number of irreducible representations equals number of conjugacy classes, element structure of projective special linear group of degree two over a finite field#Conjugacy class structure
quasirandom degree (minimum possible degree of nontrivial irreducible representation)	Case $q$ congruent to 1 mod 4 (e.g., $q=5,9,13,17,25,29$): $(q + 1)/2$ Case $q$ congruent to 3 mod 4 (e.g., $q = 3,7,11,19,23,27$): $(q - 1)/2$ Case $q$ even: $q - 1$
maximum degree of irreducible representation over a splitting field	$q + 1$
lcm of degrees of irreducible representations over a splitting field	Case $q$ odd: $q(q+1)(q-1)/2 = (q^3-q)/2$, Case $q$ even: $q(q+1)(q-1) = q^3 - q$
sum of squares of degrees of irreducible representations over a splitting field	Case $q$ odd: $(q^3 - q)/2$, case $q$ even: $q^3 - q$ equal to the group order. See sum of squares of degrees of irreducible representations equals group order