We will prove that if M is an n-dimensional smooth compact
connected manifold, of cup length n, then there exists some
constant c such that: (1) any finite group of diffeomorphisms of M
has an abelian subgroup of index at most c, (2) if the Euler characteristic
of M is nonzero, then no finite group of diffeomorphisms of M has more
than c elements. These statements can be seen as analogues of a
classical theorem of Jordan for the diffeomorphism group of M (instead
of the group GL(n,C) as in the original theorem).