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[parent] projective special linear group (Definition)
Definition 1   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 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.

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

Theorem 1   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: $$\PSL(2,F)=\SL(2,F)/\{ \pm \operatorname{Id} \}.$$

As a consequence of the previous theorem, we obtain:

Theorem 2   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 $\mathbb{F}=\mathbb{F}_2,\mathbb{F}_3$ .

Bibliography

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




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See Also: theorems of special linear group over a finite field

Other names:  PSL

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Cross-references: finite, finite field, simple group, theorem, consequence, root of unity, scalar, origin, projective space, group, matrices, linear transformations, dimension, finite dimensional, quotient group, center, special linear group, field, vector space
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This is version 1 of projective special linear group, born on 2005-03-28.
Object id is 6912, canonical name is ProjectiveSpecialLinearGroup.
Accessed 5128 times total.

Classification:
AMS MSC20G15 (Group theory and generalizations :: Linear algebraic groups :: Linear algebraic groups over arbitrary fields)

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