PlanetMath: projective special linear group

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projective special linear group

(Definition)

Definition 1 Let be a vector space over a field and let $\operatorname{SL}(V)$ be the special linear group. Let be the center of $\operatorname{SL}(V)$ . The projective special linear group associated to is the quotient group $\operatorname{SL}(V)/Z$ and is usually denoted by $\operatorname{PSL}(V)$ .

When is a finite dimensional vector space over (of dimension ) then we write $\operatorname{PSL}(n,F)$ or $\operatorname{PSL}_n(F)$ . We also identify the linear transformations of 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 of $\operatorname{SL}(n,F)$ is the group of all scalar matrices $\lambda\cdot \operatorname{Id}$ where $\lambda$ is an th root of unity in .

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

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

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

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

are all finite simple groups, except for and $\mathbb{F}=\mathbb{F}_2,\mathbb{F}_3$ .

Bibliography

1: S. Lang, Algebra, Springer-Verlag, New York.
2: D. Dummit, R. Foote, Abstract Algebra, Second Edition, Wiley.

"projective special linear group" is owned by alozano.

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PSL

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Cross-references: finite, finite field, simple group, consequence, root of unity, scalar, origin, projective space, group, matrices, linear transformations, dimension, finite dimensional, quotient group, center, special linear group, field, vector space
There are 3 references to this entry.

This is version 1 of projective special linear group, born on 2005-03-28.
Object id is 6912, canonical name is ProjectiveSpecialLinearGroup.
Accessed 3537 times total.

Classification:

AMS MSC:

20G15 (Group theory and generalizations :: Linear algebraic groups :: Linear algebraic groups over arbitrary fields)

Pending Errata and Addenda

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