On finite subgroups of compact Lie groups and fundamental groups of Riemannian manifolds

Let G be a compact Lie group of dimension m, and let Gamma be any finite subgroup in G. The main result in this paper is that for each m there exists a constant c (m) such that: (i) for connected G, Gamma contains an abelian subgroup with index <= c (m) which belongs to some torus in G; (ii) for non-connected G, Gamma contains a subgroup with index <= c (m) which commutes with some torus in G. Using this we get some conclusions on fundamental groups of Riemannian manifolds (especially of positive sectional curvature).