Projective general linear group: Difference between revisions

Latest revision as of 00:41, 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.

Arithmetic functions

Over a finite field

Below is information for the projective general linear group of degree $n$ over a finite field of size $q$ .

Function	Value	Similar groups	Explanation
order	$q^{(\binom{n}{2})} \prod_{i = 2}^{n} (q^{i} - 1)$	$S L (n, q)$ has the same order	The order of $G L (n, q)$ is $q^{(\binom{n}{2})} \prod_{i = 1}^{n} (q^{i} - 1)$ . The center has order $q - 1$ , so the quotient has order equal to the quotient of the order of $G L (n, q)$ by $q - 1$ .
number of conjugacy classes	(no good generic expression, but overall it's a polynomial in $q$ of degree $n - 1$ )		See element structure of projective general linear group over a finite field

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

If $q = 2$ , then $P G L (n, q) = G L (n, q) = S L (n, q) = P S L (n, q)$ for all $n$ .

More generally, if $q - 1$ is relatively prime to $n$ , then the groups $P G L (n, q), S L (n, q), P S L (n, q)$ are all isomorphic to each other. However, they are not isomorphic to $G L (n, q)$ .

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 $P G L (n, q) = P G L (n, F_{q})$	Order of $P G L (n, q)$
$q$	$p$	1	Trivial group	1
2	2	2	Symmetric group:S3	6
3	3	2	Symmetric group:S4	24
4	2	2	Alternating group:A5	60
5	5	2	Symmetric group:S5	120
9	3	2	Projective general linear group:PGL(2,9)	720
2	2	3	Projective special linear group:PSL(3,2)	168

@@ Line 18: / Line 18: @@
 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.
+==Arithmetic functions==
+===Over a finite field===
+Below is information for the projective general linear group of degree <math>n</math> over a finite field of size <math>q</math>.
+{| class="sortable" border="1"
+! Function !! Value !! Similar groups !! Explanation
+|-
+| [[order of a group|order]] || <math>q^{\binom{n}{2}}\prod_{i=2}^n (q^i - 1)</math> || <math>SL(n,q)</math> has the same order || The order of <math>GL(n,q)</math> is <math>q^{\binom{n}{2}}\prod_{i=1}^n (q^i - 1)</math>. The center has order <math>q - 1</math>, so the quotient has order equal to the quotient of the order of <math>GL(n,q)</math> by <math>q - 1</math>.
+|-
+| [[number of conjugacy classes]] || (no good generic expression, but overall it's a polynomial in <math>q</math> of degree <math>n - 1</math>) || || See [[element structure of projective general linear group over a finite field]]
+|}
 ==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>.
+If <math>q = 2</math>, then <math>PGL(n,q) = GL(n,q) = SL(n,q) = PSL(n,q)</math> for all <math>n</math>.
+More generally, if <math>q - 1</math> is relatively prime to <math>n</math>, then the groups <math>PGL(n,q), SL(n,q), PSL(n,q)</math> are all isomorphic to each other. However, they are not isomorphic to <math>GL(n,q)</math>.
-{| class="wikitable" border="1"
+{| class="sortable" border="1"
-!Size of field !! Order of matrices !! Common name for the projective special linear group
+!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> !! Order of <math>PGL(n,q)</math>
 |-
-| <math>q</math> || 1 || [[Trivial group]]
+| <math>q</math> || <math>p</math> || 1 || [[Trivial group]] || 1
 |-
-| 2 || 2 || [[Symmetric group:S3]]
+| 2 || 2 ||2 || [[Symmetric group:S3]] || 6
 |-
-| 3 || 2 || [[Symmetric group:S4]]
+| 3 || 3 || 2 || [[Symmetric group:S4]] || 24
 |-
-| 4 || 2 || [[Alternating group:A5]]
+| 4 || 2 || 2 || [[Alternating group:A5]] || 60
 |-
-| 5 || 2 || [[Symmetric group:S5]]
+| 5 || 5 || 2 || [[Symmetric group:S5]] || 120
 |-
-| 9 || 2 || [[Projective general linear group:PGL(2,9)]]
+| 9 || 3 || 2 || [[Projective general linear group:PGL(2,9)]] || 720
 |-
-| 2 || 3 ||[[Projective special linear group:PSL(3,2)]]
+| 2 || 2 || 3 ||[[Projective special linear group:PSL(3,2)]] || 168
 |}