  We consider the following problem: for which classes of finite groups, and in
particular finite simple groups, does the minimal dimension of a faithful,
smooth action on a homology sphere coincide with the minimal dimension of a
faithful, linear action on a sphere? We prove that the two minimal dimensions
coincide for the linear fractional groups PSL(2,p) as well as for various
classes of alternating and symmetric groups. We prove analogous results also
for actions on Euclidean spaces.
