In the two earlier papers in this
series we showed, that if the finite group π has period 2d in cohomology, and if
for all primes p a subgroup of order 2p is cyclic, then there exists a free
topological action by π on a sphere of dimension S2nd−1, for some positive
integer n. Two questions remained open, namely whether there also existed
smooth actions, and whether n could be taken equal to one. In this short
paper we prove that there exists a free smooth action of π on S2e(π)−1, the
sphere with the standard differentiable structure. Here e(π) is the Artin-Lam
induction exponent, that is, the least positive integer such that e(π)1 belongs to
the ideal of the rational representation ring, generated by representations
induced from cyclic subgroups. It turns out that e(π) = d(π) or 2d(π), and
that our result is geometrically the best possible, except for one class of
groups.
