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Algebraic groups with few subgroups
Author(s) -
Garibaldi Skip,
Gille Philippe
Publication year - 2009
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdp030
Subject(s) - mathematics , algebraic number , type (biology) , algebraic group , torus , field (mathematics) , pure mathematics , group (periodic table) , combinatorics , discrete mathematics , physics , mathematical analysis , geometry , ecology , quantum mechanics , biology
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 E 8 have no others. We give a simpler proof of his result, prove that some groups of type 3 D 4 and 6 D 4 have no nontrivial connected subgroups, and give partial results for types E 6 and E 7 . Our result for 3 D 4 uses a general theorem on the indexes of Tits algebras that is of independent interest.