Finite isometry groups of 4-manifolds with positive sectional curvature

Let M be an oriented compact positively curved 4-manifold. Let G be a finite subgroup of the isometry group of $M$. Among others, we prove that there is a universal constant C (cf. Corollary 4.3 for the approximate value of C), such that if the order of G is odd and at least C, then G is either abelian of rank at most 2, or non-abelian and isomorphic to a subgroup of PU(3) with a presentation \{A, B| A^m=B^n=1, BAB^{-1}=A^r, (n(r-1), m)=1, r\ne r^3=1(\text{mod}m) \}. Moreover, M is homeomorphic to CP^2 if G is non-abelian, and homeomorphic to S^4 or CP^2 if G is abelian of rank 2.