projective special linear group

Definition.

Let $V$ be a vector space over a field $F$ and let $\operatorname{SL}(V)$ be the special linear group. Let $Z$ be the center of $\operatorname{SL}(V)$. The associated to $V$ is the quotient group $\operatorname{SL}(V)/Z$ and is usually denoted by $\operatorname{PSL}(V)$.

When $V$ is a finite dimensional vector space over $F$ (of dimension $n$) then we write $\operatorname{PSL}(n,F)$ or $\operatorname{PSL}_{n}(F)$. We also identify the linear transformations of $V$ with $n\times n$ matrices, so $\operatorname{PSL}$ may be regarded as a quotient of the group of matrices $\operatorname{SL}(n,F)$ by its center.

Note: see the entry on projective space for the origin of the terminology.

Theorem 1.

The center $Z$ of $\operatorname{SL}(n,F)$ is the group of all scalar matrices $\lambda\cdot\operatorname{Id}$ where $\lambda$ is an $n$th root of unity in $F$.

In particular, for $n=2$, $Z=\{\pm\operatorname{Id}\}$ and:

 $\operatorname{PSL}(2,F)=\operatorname{SL}(2,F)/\{\pm\operatorname{Id}\}.$

As a consequence of the previous theorem, we obtain:

Theorem 2.

For $n\geq 3$, $\operatorname{PSL}(n,F)$ is a simple group. Furthermore, if $\mathbb{F}$ is a finite field then the groups

 $\operatorname{PSL}(n,\mathbb{F})=\operatorname{SL}(n,\mathbb{F})/Z,\quad n\geq 2$

are all finite simple groups, except for $n=2$ and $\mathbb{F}=\mathbb{F}_{2},\mathbb{F}_{3}$.

References

• 1 S. Lang, , Springer-Verlag, New York.
• 2 D. Dummit, R. Foote, Abstract Algebra, Second Edition, Wiley.
Title projective special linear group ProjectiveSpecialLinearGroup 2013-03-22 15:09:46 2013-03-22 15:09:46 alozano (2414) alozano (2414) 4 alozano (2414) Definition msc 20G15 PSL TheoremsOfSpecialLinearGroupOverAFiniteField