ON THE COMPONENT GROUP OF THE AUTOMORPHISM GROUP OF A LIE GROUP

Let G be a Lie group, G-degree the connected component of G that conta ins the identity, and Aut G the group of all topological automorphisms of G. In the case when G/G-degree is finite and G has a faithful repr esentation, we obtain a necessary and sufficient condition for G so th at Aut G has finitely many components in terms of the maximal central torus in G-degree.