  We determine all the normal subgroups of the group of C^r diffeomorphisms of
R^n, r = 1,2,...,infinity, except when r=n+1 or n=4, and also of the group of
homeomorphisms of R^n (r=0). We also study the group A_0 of diffeomorphisms of
an open manifold M that are isotopic to the identity. If M is the interior of a
compact manifold with nonempty boundary, then the quotient of A_0 by the normal
subgroup of diffeomorphisms that coincide with the identity near to a given end
e of M is simple.
