  We give a lower bound for the exponent of certain elements in the integral
cohomology of the total spaces of principal BC-bundles for C a finite cyclic
group. As applications we give a proof of the theorem of A. Adem and H.-W. Henn
that a p-group is elementary abelian if and only if its integral cohomology has
exponent p, and we exhibit some infinite groups of finite virtual cohomological
dimension whose Tate-Farrell cohomology contains torsion of order greater than
the l.c.m. of the orders of their finite subgroups. We also give an upper bound
for the exponent of all but finitely many of the integral cohomology groups of
a finite group, in terms of the permutation representations of the group.
