Actually, a Lie Group is a group which is simultaneously a manifold, such that the group multiplication corresponds to diffeomorphisms of the manifold onto itse

Actually, a Lie Group is a group which is simultaneously a manifold, such that the group multiplication corresponds to diffeomorphisms of the manifold onto itself. The Lie Algebra corresponding to that Lie Group is the tangent space of the identity. The elements of the Lie Algebra generate the given Lie Group via the exponential map.