Effective action of finite group of diffeomorphisms has free point

Statement

Suppose $M$ is a connected differential manifold and $G$ is a finite subgroup of the self-diffeomorphism group of $M$. In other words, $G$ is a finite group acting on $M$ such that no non-identity element of $G$ acts trivially on $M$.

Then, the action of $G$ on $M$ has a free point: a point whose isotropy subgroup (stabilizer) is trivial.

Definitions used

Fill this in later

Facts used

We use two main facts:

1. Fixed-point set of finite group of diffeomorphisms is closed submanifold
2. A finite union of submanifolds of codimension at least one is proper.

Proof

For every $g\in G$ other than the identity element, consider the set $M^{g}$ of elements of $M$ fixed by the cyclic subgroup generated by $g$. By fact (1) above, this is a closed submanifold, and by the effectiveness assumption, it cannot be the whole of $M$. Since $M$ is connected, it cannot be of full dimension, otherwise it would be both closed and open. Hence $M^{g}$ is either empty, or is a submanifold of codimension at least $1$.

Now, consider:

$\bigcup _{g\in G\setminus \{e\}}M^{g}$

This is a finite union of submanifolds of codimension at least 1, so is a proper subset. Thus, we can pick a point not fixed by any $g\in G$, and this is the desired free point.