projective special linear group
Let be a vector space over a field and let be the special linear group. Let be the center of . The projective special linear group associated to is the quotient group and is usually denoted by .
When is a finite dimensional vector space over (of dimension ) then we write or . We also identify the linear transformations of with matrices, so may be regarded as a quotient of the group of matrices by its center.
Note: see the entry on projective space for the origin of the terminology.
The center of is the group of all scalar matrices where is an th root of unity in .
In particular, for , and:
As a consequence of the previous theorem, we obtain:
For , is a simple group. Furthermore, if is a finite field then the groups
are all finite simple groups, except for and .
- 1 S. Lang, Algebra, Springer-Verlag, New York.
- 2 D. Dummit, R. Foote, Abstract Algebra, Second Edition, Wiley.
|Title||projective special linear group|
|Date of creation||2013-03-22 15:09:46|
|Last modified on||2013-03-22 15:09:46|
|Last modified by||alozano (2414)|