projective special linear group
Definition.
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.
Theorem 1.
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:
Theorem 2.
For , is a simple group. Furthermore, if is a finite field then the groups
are all finite simple groups, except for and .
References
- 1 S. Lang, Algebra, Springer-Verlag, New York.
- 2 D. Dummit, R. Foote, Abstract Algebra, Second Edition, Wiley.
Title | projective special linear group |
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Canonical name | ProjectiveSpecialLinearGroup |
Date of creation | 2013-03-22 15:09:46 |
Last modified on | 2013-03-22 15:09:46 |
Owner | alozano (2414) |
Last modified by | alozano (2414) |
Numerical id | 4 |
Author | alozano (2414) |
Entry type | Definition |
Classification | msc 20G15 |
Synonym | PSL |
Related topic | TheoremsOfSpecialLinearGroupOverAFiniteField |