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Difference between revisions of "Dickson group"

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The group of exponential automorphisms of a classical simple Lie algebra of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031620/d0316201.png" /> over a finite field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031620/d0316202.png" />. If the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031620/d0316203.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031620/d0316204.png" />, the order of the Dickson group is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031620/d0316205.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031620/d0316206.png" /> the Dickson group is a simple group. These groups were discovered by L.E. Dickson [[#References|[1]]]. During the 50 years which followed no new finite simple group could be discovered, until a general method for obtaining simple groups as groups of automorphisms of simple Lie algebras was discovered by C. Chevalley [[#References|[2]]]. In particular, Chevalley's method makes it possible to obtain Dickson groups as well [[#References|[3]]].
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The group of exponential automorphisms of a classical simple Lie algebra of type $G_2$ over a finite field $F$. If the order of $F$ is $q$, the order of the Dickson group is $q^6(q^2-1)(q^6-1)$. If $q>2$ the Dickson group is a simple group. These groups were discovered by L.E. Dickson [[#References|[1]]]. During the 50 years which followed no new finite simple group could be discovered, until a general method for obtaining simple groups as groups of automorphisms of simple Lie algebras was discovered by C. Chevalley [[#References|[2]]]. In particular, Chevalley's method makes it possible to obtain Dickson groups as well [[#References|[3]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.E. Dickson,  "A new system of simple groups"  ''Math. Ann.'' , '''60'''  (1905)  pp. 137–150</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C. Chevalley,  "Sur certains groupes simples"  ''Tôhoku Math. J.'' , '''7'''  (1955)  pp. 14–66</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R.W. Carter,  "Simple groups of Lie type" , Wiley (Interscience)  (1972)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.E. Dickson,  "A new system of simple groups"  ''Math. Ann.'' , '''60'''  (1905)  pp. 137–150</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C. Chevalley,  "Sur certains groupes simples"  ''Tôhoku Math. J.'' , '''7'''  (1955)  pp. 14–66</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R.W. Carter,  "Simple groups of Lie type" , Wiley (Interscience)  (1972)</TD></TR></table>

Revision as of 19:04, 12 April 2014

The group of exponential automorphisms of a classical simple Lie algebra of type $G_2$ over a finite field $F$. If the order of $F$ is $q$, the order of the Dickson group is $q^6(q^2-1)(q^6-1)$. If $q>2$ the Dickson group is a simple group. These groups were discovered by L.E. Dickson [1]. During the 50 years which followed no new finite simple group could be discovered, until a general method for obtaining simple groups as groups of automorphisms of simple Lie algebras was discovered by C. Chevalley [2]. In particular, Chevalley's method makes it possible to obtain Dickson groups as well [3].

References

[1] L.E. Dickson, "A new system of simple groups" Math. Ann. , 60 (1905) pp. 137–150
[2] C. Chevalley, "Sur certains groupes simples" Tôhoku Math. J. , 7 (1955) pp. 14–66
[3] R.W. Carter, "Simple groups of Lie type" , Wiley (Interscience) (1972)
How to Cite This Entry:
Dickson group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dickson_group&oldid=31648
This article was adapted from an original article by V.D. Mazurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article