Projective special linear group
From Groupprops
Particular cases
Finite fields
Some facts:
- For
,
. For
a power of two,
but this is not equal to
.
- 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.
Size of field | Order of matrices | Common name for the projective special linear group | Order of group | Comment |
---|---|---|---|---|
![]() |
1 | Trivial group | ![]() |
Trivial |
2 | 2 | Symmetric group:S3 | ![]() |
supersolvable but not nilpotent. Not simple. |
3 | 2 | Alternating group:A4 | ![]() |
solvable but not supersolvable group. Not simple. |
4 | 2 | Alternating group:A5 | ![]() |
simple non-abelian group of smallest order. |
5 | 2 | Alternating group:A5 | ![]() |
simple non-abelian group of smallest order. |
7 | 2 | Projective special linear group:PSL(3,2) | ![]() |
simple non-abelian group of second smallest order. |
9 | 2 | Alternating group:A6 | ![]() |
simple non-abelian group. |
2 | 3 | Projective special linear group:PSL(3,2) | ![]() |
simple non-abelian group of second smallest order. |
3 | 3 | Projective special linear group:PSL(3,3) | ![]() |
simple non-abelian group. |