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''in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075250/p0752501.png" /> variables over a [[Skew-field|skew-field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075250/p0752502.png" />''
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{{MSC|20}}
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{{TEX|done}}
  
The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075250/p0752503.png" /> of transformations of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075250/p0752504.png" />-dimensional [[Projective space|projective space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075250/p0752505.png" /> induced by the linear transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075250/p0752506.png" />. There is a natural [[Epimorphism|epimorphism]]
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''in $n$ variables over a
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[[Skew-field|skew-field]] $K$''
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075250/p0752507.png" /></td> </tr></table>
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The group $\def\PGL{ {\rm PGL}}\PGL_n(K)$ of transformations of the $(n-1)$-dimensional
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[[Projective space|projective space]] $P^{n-1}(K)$ induced by the linear transformations of $K^n$. There is a natural
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[[Epimorphism|epimorphism]]
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$$P: {\rm GL}_n(K)\to \PGL_n(K),$$
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with as kernel the group of homotheties (cf.
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[[Homothety|Homothety]]) of $K^n$, which is isomorphic to the multiplicative group $Z^*$ of the centre $Z$ of $K$. The elements of $\PGL_n(K)$, called projective transformations, are the collineations (cf.
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[[Collineation|Collineation]]) of $P^{n-1}(K)$. Along with
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$\PGL_n(K)$, which is also called the full projective group, one also
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considers the unimodular projective group $\def\PSL{ {\rm
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PSL}}\PSL_n(K)$, and, in general, groups of the form $P(G) \subset \PGL_n(K)$, where $G
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\subset {\rm GL}_n(K)$ is a
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[[Linear group|linear group]].
  
with as kernel the group of homotheties (cf. [[Homothety|Homothety]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075250/p0752508.png" />, which is isomorphic to the multiplicative group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075250/p0752509.png" /> of the centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075250/p07525010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075250/p07525011.png" />. The elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075250/p07525012.png" />, called projective transformations, are the collineations (cf. [[Collineation|Collineation]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075250/p07525013.png" />. Along with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075250/p07525014.png" />, which is also called the full projective group, one also considers the unimodular projective group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075250/p07525015.png" />, and, in general, groups of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075250/p07525016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075250/p07525017.png" /> is a [[Linear group|linear group]].
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For $n\ge 2$ the group $\PSL_n(K)$ is simple, except for the two cases $n=2$ and $|K|=2$ or 3. If $K$ is the finite field of $q$ elements, then
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075250/p07525018.png" /> the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075250/p07525019.png" /> is simple, except for the two cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075250/p07525020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075250/p07525021.png" /> or 3. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075250/p07525022.png" /> is the finite field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075250/p07525023.png" /> elements, then
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$$|\PSL_n(K)| = (q-1,n)^{-1} q^{n(n-1)/2} (q^n-1)(q^{n-1}-1)\cdots (q^2-1).$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075250/p07525024.png" /></td> </tr></table>
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For a brief resumé on the orders of the other finite classical groups, like ${\rm PSp}_n$, and their simplicity cf. e.g.
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{{Cite|Ca}}.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075250/p07525025.png" /></td> </tr></table>
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.A. Dieudonné, "La géométrie des groupes classiques" , Springer (1955) {{MR|0072144}} {{ZBL|0067.26104}} </TD></TR></table>
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{|
 
+
|-
 
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|valign="top"|{{Ref|Ca}}||valign="top"| R.W. Carter, "Simple groups of Lie type", Wiley (Interscience) (1972) pp. Chapt. 1 {{MR|0407163}} {{ZBL|0248.20015}}
 
+
|-
====Comments====
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|valign="top"|{{Ref|Di}}||valign="top"| J.A. Dieudonné, "La géométrie des groupes classiques", Springer (1955) {{MR|0072144}} {{ZBL|0067.26104}} 
The groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075250/p07525026.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075250/p07525027.png" /> are the images of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075250/p07525028.png" /> under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075250/p07525029.png" />. For a brief resumé on the orders of the other finite classical groups, like <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075250/p07525030.png" />, and their simplicity cf. e.g. [[#References|[a1]]].
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|-
 
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|}
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R.W. Carter, "Simple groups of Lie type" , Wiley (Interscience) (1972) pp. Chapt. 1</TD></TR></table>
 

Latest revision as of 16:49, 3 November 2013

2010 Mathematics Subject Classification: Primary: 20-XX [MSN][ZBL]

in $n$ variables over a skew-field $K$

The group $\def\PGL{ {\rm PGL}}\PGL_n(K)$ of transformations of the $(n-1)$-dimensional projective space $P^{n-1}(K)$ induced by the linear transformations of $K^n$. There is a natural epimorphism $$P: {\rm GL}_n(K)\to \PGL_n(K),$$ with as kernel the group of homotheties (cf. Homothety) of $K^n$, which is isomorphic to the multiplicative group $Z^*$ of the centre $Z$ of $K$. The elements of $\PGL_n(K)$, called projective transformations, are the collineations (cf. Collineation) of $P^{n-1}(K)$. Along with $\PGL_n(K)$, which is also called the full projective group, one also considers the unimodular projective group $\def\PSL{ {\rm PSL}}\PSL_n(K)$, and, in general, groups of the form $P(G) \subset \PGL_n(K)$, where $G \subset {\rm GL}_n(K)$ is a linear group.

For $n\ge 2$ the group $\PSL_n(K)$ is simple, except for the two cases $n=2$ and $|K|=2$ or 3. If $K$ is the finite field of $q$ elements, then

$$|\PSL_n(K)| = (q-1,n)^{-1} q^{n(n-1)/2} (q^n-1)(q^{n-1}-1)\cdots (q^2-1).$$

For a brief resumé on the orders of the other finite classical groups, like ${\rm PSp}_n$, and their simplicity cf. e.g. [Ca].


References

[Ca] R.W. Carter, "Simple groups of Lie type", Wiley (Interscience) (1972) pp. Chapt. 1 MR0407163 Zbl 0248.20015
[Di] J.A. Dieudonné, "La géométrie des groupes classiques", Springer (1955) MR0072144 Zbl 0067.26104
How to Cite This Entry:
Projective group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Projective_group&oldid=24163
This article was adapted from an original article by E.B. Vinberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article