Algebraic groups with few subgroups

Abstract

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.