# 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

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

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