projective special linear group

Definition.

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

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

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

Theorem 1.

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

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

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

As a consequence of the previous theorem, we obtain:

Theorem 2.

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

$$\mathrm{PSL}(n,𝔽)=\mathrm{SL}(n,𝔽)/Z,n\ge 2$$

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