Difference between revisions of "Dickson group"
From Encyclopedia of Mathematics
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| − | The group of exponential automorphisms of a classical simple Lie algebra of type | + | {{TEX|done}} |
| + | 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
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