Positive sectional curvature, symmetry and Chern's conjecture

Let M be a n-manifold of positive sectional curvature. Suppose that M admits an isometrical torus Tk-action with k>n+18+1. The main results of the paper are: (1) the fundamental group π1(M) contains no Zp⊕Zp subgroup with p prime and p≠3 (a partial positive answer to Chern's conjecture); (2) the 2-order element of π1(M) belongs to the center of π1(M).