Abstract Let G be a compact Lie group of dimension m, and let Γ 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, Γ contains an abelian subgroup with index ≤ c (m) which belongs to some torus in G; (ii) for non-connected G, Γ 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).