Projective special linear group

Particular cases

Finite fields

Some facts:

• For $q=2$, $PSL(n,q)=SL(n,q)=PGL(n,q)=GL(n,q)$. For $q$ a power of two, $PSL(n,q)=SL(n,q)$ but this is not equal to $GL(n,q)$.
• Projective special linear group equals alternating group in only finitely many cases: All those cases are listed in the table below.
• Projective special linear group is simple except for finitely many cases, all of which are listed below.
• The projective special linear group is isomorphic to the quotient group of the special linear group by its center. In symbols, $PSL(n,q)\cong SL(n,q)/Z(SL(n,q))$.

Size of field	Order of matrices	Common name for the projective special linear group	Order of group	Comment
$q$	1	Trivial group	$1$	Trivial
2	2	Symmetric group:S3	$6=2\cdot 3$	supersolvable but not nilpotent. Not simple.
3	2	Alternating group:A4	$12=2^{2}\cdot 3$	solvable but not supersolvable group. Not simple.
4	2	Alternating group:A5	$60=2^{2}\cdot 3\cdot 5$	simple non-abelian group of smallest order.
5	2	Alternating group:A5	$60=2^{2}\cdot 3\cdot 5$	simple non-abelian group of smallest order.
7	2	Projective special linear group:PSL(3,2)	$168=2^{3}\cdot 3\cdot 7$	simple non-abelian group of second smallest order.
9	2	Alternating group:A6	$360=2^{3}\cdot 3^{2}\cdot 5$	simple non-abelian group.
2	3	Projective special linear group:PSL(3,2)	$168=2^{3}\cdot 3\cdot 7$	simple non-abelian group of second smallest order.
3	3	Projective special linear group:PSL(3,3)	$5616=2^{4}\cdot 3^{3}\cdot 13$	simple non-abelian group.

More information

• Element structure of projective special linear group over a finite field
• Linear representation theory of projective special linear group over a finite field