  Let M be a closed simply connected n-manifold of positive sectional
curvature. We determine its homeomorphism or homotopic type if M also admits an
isometric elementary p-group action of large rank. Our main results are: There
exists a constant p(n)>0 such that (1) If M^{2n} admits an effective isometric
\Bbb Z_p^k-action for a prime p\ge p(n), then k\le n and ``='' implies that
M^{2n} is homeomorphic to a sphere or a complex projective space. (2) If
M^{2n+1} admits an isometric S^1 x \Bbb Z_p^k-action for a prime p\ge p(n),
then k\le n and ``='' implies that M is homeomorphic to a sphere. (3) For M in
(1) or (2), if n\ge 7 and k\ge [\frac{3n}4]+2, then M is homeomorphic to a
sphere or homotopic to a complex projective space.
