Compact Lie groups have a very well understood structure theory.
All maximal tori of a compact Lie group are conjugate by inner automorphisms. The dimension of a maximal torus of a compact Lie group is called a rank of . The normalizer of a maximal torus determines . The Weyl group of with respect to a choice of a maximal torus is the group of automorphisms of which are restrictions of inner automorphisms of . The maximal torus is of finite index in its normalizer; the quotient is isomorphic to . The cardinality of for a compact connected , equals the Euler characteristic of the homogeneous space (“flag variety”).