Projective general linear group: Difference between revisions

Revision as of 00:32, 7 November 2011

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 $k^{*}$ )

For $q$ a prime power, we denote by $P G L (n, q)$ the group $P G L (n, F_{q})$ where $F_{q}$ is the field (unique up to isomorphism) of size $q$ .

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

Particular cases by degree

Degree $n$	Information on projective general linear group $P G L (n, k)$ over a field $k$
1	trivial group always
2	projective general linear group of degree two
3	projective general linear group of degree three
4	projective general linear group of degree four

Finite fields

Size of field $q$	Characteristic $p$ of field (so $q$ is a power of $p$	Degree of projective general linear group $n$	Common name for the projective general linear group $P G L (n, q) = P G L (n, F_{q})$
$q$	$p$	1	Trivial group
2	2	2	Symmetric group:S3
3	3	2	Symmetric group:S4
4	2	2	Alternating group:A5
5	5	2	Symmetric group:S5
9	3	2	Projective general linear group:PGL(2,9)
2	2	3	Projective special linear group:PSL(3,2)

@@ Line 20: / Line 20: @@
 ==Particular cases==
+===Particular cases by degree===
+{| class="sortable" border="1"
+! Degree <math>n</math> !! Information on projective general linear group <math>PGL(n,k)</math> over a field <math>k</math>
+|-
+| 1 || [[trivial group]] always
+|-
+| 2 || [[projective general linear group of degree two]]
+|-
+| 3 || [[projective general linear group of degree three]]
+|-
+| 4 || [[projective general linear group of degree four]]
+|}
 ===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="sortable" border="1"
+!Size of field <math>q</math> !! Characteristic <math>p</math> of field (so <math>q</math> is a power of <math>p</math> !! Degree of projective general linear group <math>n</math> !! Common name for the projective general linear group <math>PGL(n,q) = PGL(n,\mathbb{F}_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]]
+| <math>q</math> || <math>p</math> || 1 || [[Trivial group]]
 |-
-| 2 || 2 || [[Symmetric group:S3]]
+| 2 || 2 ||2 || [[Symmetric group:S3]]
 |-
-| 3 || 2 || [[Symmetric group:S4]]
+| 3 || 3 || 2 || [[Symmetric group:S4]]
 |-
-| 4 || 2 || [[Alternating group:A5]]
+| 4 || 2 || 2 || [[Alternating group:A5]]
 |-
-| 5 || 2 || [[Symmetric group:S5]]
+| 5 || 5 || 2 || [[Symmetric group:S5]]
 |-
-| 9 || 2 || [[Projective general linear group:PGL(2,9)]]
+| 9 || 3 || 2 || [[Projective general linear group:PGL(2,9)]]
 |-
-| 2 || 3 ||[[Projective special linear group:PSL(3,2)]]
+| 2 || 2 || 3 ||[[Projective special linear group:PSL(3,2)]]
 |}