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 $P S L (2, R)$ or $P S L_{2} (R)$ .

For a prime power

Suppose $q$ is a prime power. The projective special linear group $P S L (2, q)$ 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 $q$

Value of prime power $q$	Underlying prime $p$	Exponent on $p$ giving $q$	Group $P S L (2, q)$	Order: $q (q + 1) (q - 1) / g c d (2, q - 1)$	Second part of GAP ID (if applicable)	Comments
2	2	1	symmetric group:S3	6	1	not simple (one of two exceptions)
3	3	1	alternating group:A4	12	3	not simple (one of two exceptions)
4	2	2	alternating group:A5	60	5	minimal simple group
5	5	1	alternating group:A5	60	5	minimal simple group
7	7	1	projective special linear group:PSL(3,2)	168	42	minimal simple group
8	2	3	projective special linear group:PSL(2,8)	504	156	minimal simple group
9	3	2	alternating group:A6	360	114	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 $P S L (2, 4)$
17	17	1	projective special linear group:PSL(2,17)	2448	--	minimal simple group

Arithmetic functions

Below we give the arithmetic functions for $P S L (2, q)$ , where $q$ is a power $p^{r}$ of a prime $p$ .

Function	Value	Explanation
order	$\frac{q^{3} - q}{g c d (2, q - 1)} = \frac{q (q - 1) (q + 1)}{g c d (2, q - 1)}$ Becomes $(q^{3} - q) / 2$ for odd $q$ , $q^{3} - q$ when $p = 2$	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page. . 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 $P S L (2, q)$ , $q$ 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 $P S L (2, 5) ≅ P S L (2, 4)$ 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

Further information: element structure of projective special linear group of degree two