Let M be a closed, orientable 3-dimensional differentiable manifold, and let G be a finite group of orientation preserving diffeomorphisms of M. Let P and Q denote the set of those points of M whose stabilizer is nontrivial (that is, contains a nonidentity element of G) and noncyclic, respectively. Let χ(P) denote the Euler characteristic of P. Prove that the order of G divides χ(P), and Q is the union of −2∣G∣χ(P) orbits of G. college contestsMiklos Schweitzertopologymanifolds