Note on Lie algebras , finite groups and finite geometries
نویسنده
چکیده
The subject of this note began with Thompson [Thl,2]. In the course of constructing his simple group Th, he considered the Lie algebra Lover iC of type Es. He constructed a decomposition L H1 ..L ... ..L H31 using a family H. {H1, ... , H31 } of Cartan algebras that are pairwise perpendicular with respect to the Killing form (he called this a "Dempwolff decomposition" of L; the construction was m~de with the assistance of P. Smith and a computer). Moreover, H. is dosed under Lie multiplication. Thompson showed that such a decomposition is unique up to conjugacy in Es(iC). Moreover, he found that the stabili>ler of H. in Es(iC) has a subgroup that is a nonsplit extension of the form 25 SL(3, 2), and he constructed a lattice A C L, invariant under this group and positive definite with respect to the Killing form, such that Aut A 22 X Th.
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