CHARACTERIZING SEMISIMPLE LIE GROUPS BY CERTAIN FINITE SUBGROUPS
OROSZ, LUIZ PEDRO
Doctor of Philosophy
This thesis is concerned with a certain class of finite subgroups of Lie groups. Let G be a compact connected simply connected simple Lie group, T be a maximal torus in G and N be the normalizer of T in G. Then we have a topological group extension i p (1) O (--->) T (--->) N (--->) W (--->) 1, with W = N/T being the Weyl group. The Weyl group is a finite group which, while it gives information about G does not characterize G. (We can have isomorphic Weyl groups for nonisomorphic groups.) We are concerned with a finite group extension sitting inside (1); namely i p (2) O (--->) F (--->) (')J(G) (--->) W (--->) 1. Here F is the group of all fourth roots of unity in T and (')J(G) is a finite subgroup of N (which is an extension of a subgroup J(G) of N defined by J. Tits). It is known that if the (')J(G) groups are isomorphic for two Lie groups then the Lie groups are isomorphic. We seek a group-theoretic characterization of those finite groups which arise as (')J(G) for some Lie group G (compact, connected, simply-connected and simple). A really good characterization would allow one to prove the Cartan-Killing classification as a theorem in finite group theory. We know presentations for all (')J(G) and from these we calculate their abelianizations. Given a group extension (2), the set of all other extensions with the same F and W and the same action (phi) of W on F form a cohomology group H(,(phi))('2)(W,F). We study how the presentation of (')J(G) may be modified to give other extensions in this cohomology class, and obtain some fairly definitive results along these lines. Some preliminary results are computed for (')J(G) where G is an exceptional Lie group.