projective special linear groupProjectiveSpecialLinearGroup2005-03-28 11:36:172005-03-28 11:36:17Definitionspecial linear group% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\PSL}{\operatorname{PSL}}\begin{defn}
Let $V$ be a vector space over a field $F$ and let $\SL(V)$ be the special linear group. Let $Z$ be the center of $\SL(V)$. The {\bf projective special linear group} associated to $V$ is the quotient group $\SL(V)/Z$ and is usually denoted by $\PSL(V)$.
When $V$ is a finite dimensional vector space over $F$ (of dimension $n$) then we write $\PSL(n,F)$ or $\PSL_n(F)$. We also identify the linear transformations of $V$ with $n\times n$ matrices, so $\PSL$ may be regarded as a quotient of the group of matrices $\SL(n,F)$ by its center.
\end{defn}
Note: see the entry on projective space for the origin of the terminology.
\begin{thm}
The center $Z$ of $\SL(n,F)$ 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:
$$\PSL(2,F)=\SL(2,F)/\{ \pm \operatorname{Id} \}.$$
\end{thm}
As a consequence of the previous theorem, we obtain:
\begin{thm}
For $n\geq 3$, $\PSL(n,F)$ is a simple group.
Furthermore, if $\mathbb{F}$ is a finite field then the groups
$$\PSL(n,\mathbb{F})=\SL(n,\mathbb{F})/Z,\quad n\geq 2$$
are all finite simple groups, except for $n=2$ and $\mathbb{F}=\mathbb{F}_2,\mathbb{F}_3$.
\end{thm}
\begin{thebibliography}{00}
\bibitem{lang} S. Lang, {\em Algebra}, Springer-Verlag, New York.
\bibitem{dummit} D. Dummit, R. Foote, {\em Abstract Algebra},
Second Edition, Wiley.
\end{thebibliography}