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Let M be a closed n-manifold of positive sectional curvature.Assume that M admits an effective isometrical T~1 × Z_p~k-action with p prime.The main result of the article is that if k = 1 for n = 3 or k >n+1/4 for n ≥ 5,then there exists a positive constant p(n),depending only on n,such that π_1(M) is cyclic if p ≥ p(n).