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Journal of the London Mathematical Society Advance Access originally published online on August 4, 2009
Journal of the London Mathematical Society 2009 80(2):405-430; doi:10.1112/jlms/jdp030
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© 2009 London Mathematical Society

Algebraic groups with few subgroups

Skip Garibaldi

Department of Mathematics and Computer Science
Emory University
Atlanta, GA 30322
USA
skip@member.ams.org

Philippe Gille

UMR 8552 du CNRS
DMA - École normale supérieure
F-75005 Paris
France

Every semisimple linear algebraic group over a field F contains nontrivial connected subgroups, namely, maximal tori. In the early 1990s, J. Tits proved that some groups of type E8 have no others. We give a simpler proof of his result, prove that some groups of type 3D4 and 6D4 have no nontrivial connected subgroups, and give partial results for types E6 and E7. Our result for 3D4 uses a general theorem on the indexes of Tits algebras that is of independent interest.

2000 Mathematics Subject Classification 20G15.

The first author was partially supported by National Science Foundation grant DMS-0653502.

Received June 7, 2008; revised January 9, 2009; published online August 4, 2009.


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