Let $G$ be a Lie group and $M$ be a $C^\infty$ manifold. My textbook defines a differentiable action of $G$ on $M$ as a map $$G\times M\longrightarrow M, (g, p)\longmapsto g\cdot p,$$ such that:
(i) For all $g\in G$ the map $$L_g:M\longrightarrow M, p\longmapsto g\cdot p,$$ is a diffeomorphism.
(ii) $g\cdot (h\cdot p)=(gh)\cdot p$ for all $g, h\in G$ and $p\in M$.
I have a few questions as to this definition. I was expecting the condition $$e\cdot p=p,\ \forall p\in M,$$ where $e$ is the identity of $G$.
Indeed, if $L_g$ is only $C^\infty$ in $(i)$ and the above condition holds then $L_g$ ia a diffeomorphism.
However, if $L_g$ is a diffeomorphism as $(i)$ then I can't conclude $e$ acts trivially on $M$, can I?
Furtheremore, the smoothness of $(g, p)\longmapsto g\cdot p$ isn't enough to ensure the smoothness of $L_g$?