# Faithful finite group action on a manifold with one fixed point

Let a finite group $G$ act smoothly on a smooth manifold $M$ with one fixed point $p$. Then, by the Slice theorem, we are able to find a $G$-equivariant neighborhood, $U_p$, of $p$ which is $G$-diffeomorphic to $T_pM$ (it is endowed with an $\mathbb{R}G$-module structure arising from group action and we can tell about $G$ acting on it). Is it true that $T_pM$ is a faithful $\mathbb{R}G$-module? Ore, to put it differently since we have the $G$-diffeomorphism from $U_p$ to $T_pM$, is the action of $G$ on $U_p$ faithful?

Yes, this follows from the existence of a $G$-invariant Riemannian metric on $M$: if an isometry $\phi$ of a connected Riemannian manifold acts trivially on some $T_pM$ then $\phi=id$. (You prove this by looking at the exponential map.)