Let $M$ be a homogeneous space for a Lie group $G$. Then given any point $p \in M$, we have $M \cong G/H$ where $H$ is the isotropy group of $p$ ($H:= \{h \in G: hp = p\}$). It follows that there is a group homomorphism $\rho:H\to GL(T_pM)$ given by $\rho(h):= (h_*)_p$.
I came across the following paraphrased statement without proof in "Metric structures in differential geometry" by Walschap:
"It is not difficult to show that if $M$ is connected and $G$ acts faithfully on $M$, then $\rho$ is injective and thus defines a faithful representation of $H$."
Question: Why is this true? I'm been stuck proving this.
What I've tried: I know that if $M$ were a connected Riemannian homogeneous space ($G$ acts by isometries) with $G$ acting faithfully, then the isotropy representation is always faithful. This is because if $\exists p \in M$ and $h \in G$ with $hp = p$ and $(h_*)_p = id_{T_pM}$, then $(q \mapsto hq) = id_M$, and since the action is faithful then $h = e$. The proof is an open-closed argument using the fact that isometries map geodesics to geodesics.
I'm not sure if it is possible to adapt this argument. I was hoping I could replace "geodesics" by "integral curves of the infinitesimal generators of the group action", but I haven't managed to get anywhere.
Any hints/solutions would be greatly appreciated.