Since the left hand side is surjective (both exp and p are), the right hand side is surjective and hence exp : p g l ( 2 , C ) → → PGL ( 2 , C ) is surjective. Finally, recycle the argument once more, but now with the known isomorphism between SO(3; 1) and PGL ( 2 , C ) to find that exp is onto for the connected component of the Lorentz group.