Corollary

Let G be a topological group, N a discrete normal subgroup of G and \pi the canonical mapping of G onto G/N. Suppose that an analytic manifold structure is given on G/N, compatible with the topological group structure on G/N. Then there exists a unique manifold structure on G such that \pi is an immersion; this structure is the inverse image under \pi of the manifold structure on G/N. With this structure, \pi is étale, G is a Lie group and G/N is the quotient Lie group of G by N.