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 M2n admits an effective isometric ℤpk-action for a prime p ≥ p(n), then k ≤ n and "=" implies that M2n is homeomorphic to a sphere or a complex projective space. (2) If M2n+1 admits an isometric S1 ℤpk-action for a prime p ≥ p(n), then k ≤ n and "=" implies that M is homeomorphic to a sphere. (3) For M in (1) or (2), if n ≥ 7 and k ≥ [3n/4] + 2, then M is homeomorphic to a sphere or homotopic to a complex projective space.
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