  A recent preprint of Csik\'os, Pyber and Szab\'o (arXiv:1411.7524) proves
that the diffeomorphism group of $T^2\times S^2$ is not Jordan. The purpose of
this paper is to generalize the arguments of Csik\'os, Pyber and Szab\'o in
order to obtain many other examples of compact manifolds whose diffeomorphism
group fails to be Jordan. In particular we prove that for any $\epsilon>0$
there exist manifolds admitting effective actions of arbitrarily large
$p$-groups $\Gamma$ all of whose abelian subgroups have at most
$|\Gamma|^{\epsilon}$ elements. Finally, we also recover some results on
nonexistence of effective actions of compact connected semisimple Lie group on
manifolds.
