Let M be a closed n-manifold of positive sectional curvature.Assume that M admits an effiective isometrical T 1×Zkp-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 π1(M) is cyclic if p≥p(n).
