  By a classical result of Jordan, each finite subgroup G of a complex linear
group GL_n(C) has an abelian subgroup whose index in G is bounded by a constant
depending only on n. We consider the problem if this remains true for finite
subgroups G of the diffeomorphism group of a smooth manifold, and show that it
is true for all compact 3-manifolds as well as for Euclidean spaces of
dimension n < 7. The question remains open at present e.g. for odd-dimensional
spheres of dimension greater or equal to five, and for Euclidean spaces of
dimension greater or equal to seven.
