Projective general linear group - Revision history

Vipul: /* Particular cases */

2011-11-07T00:41:21Z

Particular cases

← Older revision		Revision as of 00:41, 7 November 2011
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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.		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==

Vipul: /* Finite fields */

2011-11-07T00:35:43Z

Finite fields

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	\|}		\|}
	===Finite fields===		===Finite fields===

			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="sortable" border="1"		{\| class="sortable" border="1"

Vipul: /* Finite fields */

2011-11-07T00:33:35Z

Finite fields

← Older revision		Revision as of 00:33, 7 November 2011
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	{\| class="sortable" border="1"		{\| 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>		!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> \|\| <math>p</math> \|\| 1 \|\| [[Trivial group]]		\| <math>q</math> \|\| <math>p</math> \|\| 1 \|\| [[Trivial group]] \|\| 1
	\|-		\|-
	\| 2 \|\| 2 \|\|2 \|\| [[Symmetric group:S3]]		\| 2 \|\| 2 \|\|2 \|\| [[Symmetric group:S3]] \|\| 6
	\|-		\|-
	\| 3 \|\| 3 \|\| 2 \|\| [[Symmetric group:S4]]		\| 3 \|\| 3 \|\| 2 \|\| [[Symmetric group:S4]] \|\| 24
	\|-		\|-
	\| 4 \|\| 2 \|\| 2 \|\| [[Alternating group:A5]]		\| 4 \|\| 2 \|\| 2 \|\| [[Alternating group:A5]] \|\| 60
	\|-		\|-
	\| 5 \|\| 5 \|\| 2 \|\| [[Symmetric group:S5]]		\| 5 \|\| 5 \|\| 2 \|\| [[Symmetric group:S5]] \|\| 120
	\|-		\|-
	\| 9 \|\| 3 \|\| 2 \|\| [[Projective general linear group:PGL(2,9)]]		\| 9 \|\| 3 \|\| 2 \|\| [[Projective general linear group:PGL(2,9)]] \|\| 720
	\|-		\|-
	\| 2 \|\| 2 \|\| 3 \|\|[[Projective special linear group:PSL(3,2)]]		\| 2 \|\| 2 \|\| 3 \|\|[[Projective special linear group:PSL(3,2)]] \|\| 168
	\|}		\|}

Vipul: /* Particular cases */

2011-11-07T00:32:55Z

Particular cases

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	==Particular cases==		==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===		===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)]]
	\|}		\|}

Vipul: /* Definition */

2011-11-07T00:29:05Z

Definition

← Older revision		Revision as of 00:29, 7 November 2011
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	* It is the group of automorphisms of [[projective space]] of dimension <math>n-1</math>, that arise from linear automorphisms of the vector space of dimension <math>n</math>.		* It is the group of automorphisms of [[projective space]] of dimension <math>n-1</math>, that arise from linear automorphisms of the vector space of dimension <math>n</math>.
	* It is the quotient of <math>GL(n,k)</math> by its center, viz the group of scalar multiplies of the identity (isomorphic to the group <math>k^*</math>)		* It is the quotient of <math>GL(n,k)</math> by its center, viz the group of scalar multiplies of the identity (isomorphic to the group <math>k^*</math>)

			For <math>q</math> a [[prime power]], we denote by <math>PGL(n,q)</math> the group <math>PGL(n,\mathbb{F}_q)</math> where <math>\mathbb{F}_q</math> is the field (unique up to isomorphism) of size <math>q</math>.

	===In terms of vector spaces===		===In terms of vector spaces===

Vipul at 15:01, 6 August 2009

2009-08-06T15:01:44Z

← Older revision		Revision as of 15:01, 6 August 2009
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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.		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)]]
			\|}

Vipul: 2 revisions

2008-05-08T00:02:45Z

2 revisions

← Older revision	Revision as of 00:02, 8 May 2008
(No difference)

Vipul: /* In terms of dimension */

2007-08-27T03:53:57Z

In terms of dimension

← Older revision		Revision as of 03:53, 27 August 2007
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	* It is the group of automorphisms of [[projective space]] of dimension <math>n-1</math>, that arise from linear automorphisms of the vector space of dimension <math>n</math>.		* It is the group of automorphisms of [[projective space]] of dimension <math>n-1</math>, that arise from linear automorphisms of the vector space of dimension <math>n</math>.
	* It is the quotient of <math>GL(n,k)</math> by its center, viz the group of scalar multiplies of the identity (isomorphic to the group <~~mathk~~^*</math>)		* It is the quotient of <math>GL(n,k)</math> by its center, viz the group of scalar multiplies of the identity (isomorphic to the group <math>k^*</math>)

	===In terms of vector spaces===		===In terms of vector spaces===

	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.		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.

Vipul at 03:47, 27 August 2007

2007-08-27T03:47:35Z

New page

{{field-parametrized group property}}

{{natural number-parametrized group property}}

==Definition==

===In terms of dimension===

Let <math>n</math> be a [[natural number]] and <math>k</math> be a [[field]]. The '''projective general linear group''' of order <math>n</math> over <math>k</math>, denoted <math>PGL(n,k)</math> is defined in the following equivalent ways:

* It is the group of automorphisms of [[projective space]] of dimension <math>n-1</math>, that arise from linear automorphisms of the vector space of dimension <math>n</math>.
* It is the quotient of <math>GL(n,k)</math> by its center, viz the group of scalar multiplies of the identity (isomorphic to the group <mathk^*</math>)

===In terms of vector spaces===

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.