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projective special linear group
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(Definition)
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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 $\mathbb{F}=\mathbb{F}_2,\mathbb{F}_3$ .
- 1
- S. Lang, Algebra, Springer-Verlag, New York.
- 2
- D. Dummit, R. Foote, Abstract Algebra, Second Edition, Wiley.
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"projective special linear group" is owned by alozano.
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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
There are 5 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 5128 times total.
Classification:
| AMS MSC: | 20G15 (Group theory and generalizations :: Linear algebraic groups :: Linear algebraic groups over arbitrary fields) |
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Pending Errata and Addenda
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