Projective general linear group: Difference between revisions

Revision as of 15:01, 6 August 2009

This term associates to every field, a corresponding group property. In other words, given a field, every group either has the property with respect to that field or does not have the property with respect to that field

This group property is natural number-parametrized, in other words, for every natural number, we get a corresponding group property

Definition

In terms of dimension

Let $n$ be a natural number and $k$ be a field. The projective general linear group of order $n$ over $k$ , denoted $P G L (n, k)$ is defined in the following equivalent ways:

It is the group of automorphisms of projective space of dimension $n - 1$ , that arise from linear automorphisms of the vector space of dimension $n$ .
It is the quotient of $G L (n, k)$ by its center, viz the group of scalar multiplies of the identity (isomorphic to the group $k^{*}$ )

In terms of vector spaces

Let $V$ be a vector space over a field $k$ . The projective general linear group of $V$ , denoted $P G L (V)$ , is defined as the inner automorphism group of $G L (V)$ , viz the quotient of $G L (V)$ by its center, which is the group of scalar multiples of the identity transformation.

Particular cases

Finite fields

For $q = 2$ , $P S L (n, q) = S L (n, q) = P G L (n, q) = G L (n, q)$ . For $q$ a power of two, $P G L (n, q) = P S L (n, q) = S L (n, q)$ but this is not the same as $G L (n, q)$ .

Size of field	Order of matrices	Common name for the projective special linear group
$q$	1	Trivial group
2	2	Symmetric group:S3
3	2	Symmetric group:S4
4	2	Alternating group:A5
5	2	Symmetric group:S5
9	2	Projective general linear group:PGL(2,9)
2	3	Projective special linear group:PSL(3,2)

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 Let <math>V</math> be a [[vector space]] over a field <math>k</math>. The projective general linear group of <math>V</math>, denoted <math>PGL(V)</math>, is defined as the [[inner automorphism group]] of <math>GL(V)</math>, viz the quotient of <math>GL(V)</math> by its center, which is the group of scalar multiples of the identity transformation.
+==Particular cases==
+===Finite fields===
+For <math>q = 2</math>, <math>PSL(n,q) = SL(n,q) = PGL(n,q) = GL(n,q)</math>. For <math>q</math> a power of two, <math>PGL(n,q) = PSL(n,q) = SL(n,q)</math> but this is not the same as <math>GL(n,q)</math>.
+{| class="wikitable" border="1"
+!Size of field !! Order of matrices !! Common name for the projective special linear group
+|-
+| <math>q</math> || 1 || [[Trivial group]]
+|-
+| 2 || 2 || [[Symmetric group:S3]]
+|-
+| 3 || 2 || [[Symmetric group:S4]]
+|-
+| 4 || 2 || [[Alternating group:A5]]
+|-
+| 5 || 2 || [[Symmetric group:S5]]
+|-
+| 9 || 2 || [[Projective general linear group:PGL(2,9)]]
+|-
+| 2 || 3 ||[[Projective special linear group:PSL(3,2)]]
+|}