SCHWEITZER 2000 Problem 9

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 $\chi (P)$ denote the Euler characteristic of $P$ . Prove that the order of $G$ divides $\chi (P)$ , and $Q$ is the union of $-2\frac{\chi(P)}{|G|}$ orbits of $G$ .