Exercise 24, § 6

Let G, G' be finite-dimensional real Lie groups, f a surjective morphism of G into G', N its kernel, H an integral subgroup of G and H' = f(H).

(a) Suppose that N is finite. For H' to be closed, it is necessary and sufficient that H be closed.

(b) Suppose that N is compact. If H is closed, H' is closed.