$ Aut(G) = \langle Aut(G)_x, Aut(G)_e\rangle$ in a connected and arc-transitive graph Let $G$ be a connected arc-transitive graph and $x,y\in V(G)$ such that $e:=\{x,y\} \in E(G)$. If $H=\langle Aut(G)_x, Aut(G)_e\rangle$, show that for any path $p = (x,y_1,\ldots,y_n)$ in $G$ there exists $\phi\in H$ such that $\phi(x)=y_n$ and conclude that $Aut(G)=H$.
I managed  to show that $Aut(\Gamma)_x$ is transitive on the neighbors of $x$ and that $Aut(\Gamma)_e$ is transitive on $\{x,y\}$ as a set. I am not sure though how to use these two results (given as a hint). The only thing I can think of is mapping $(x,y_1)$ to $(x,y)$ which does not seem useful anyway. Does anyone have any ideas?
 A: Define $r$ to be the automorphism sending $x$ on $y$ and $y$ on $x$. We know that $r\in H$ as $r(e)=e$.
If $v\in N(x)$, define $\psi_v$ to be the automorphism sending $x$ on itself and $y$ on $v$. We know $\psi_v\in H$ as $\psi_v$ fixes $x$.
Now, for $v\in N(x)$, define $t_v=\psi_v\circ r$. We readily see that $t_v(x)=\psi_v(y)=v$ and $t_v(y)=\psi_v(x)=x$. Also, by construction, $t_v\in H$.
We now claim that there exists automorphisms $\phi_i$ ($i=1,\dots,n$) such that


*

*$\phi_i\in H$

*$\phi_i(x)=y_i$

*$\phi_i(y)=y_{i-1}$ (we denote $x=y_0$)


First define $\phi_1=t_{y_1}$, we easily see that $\phi_1$ respects the above condition. We now proceed recursively. Once $\phi_{i-1}$ is defined, we define $\phi_i$ as $$\phi_i=\phi_{i-1}\circ t_{\phi_{i-1}^{-1}(y_i)}$$
Since $y_i$ is a neighbour of $y_{i-1}$ and $\phi_{i-1}^{-1}(y_{i-1})=x$, then $\phi_{i-1}^{-1}(y_{i})$ is a neighbour of $x$, and thus $t_{\phi_{i-1}^{-1}(y_i)}$ is a valid definition.
It is clear as both automorphisms are in $H$ that $\phi_i\in H$. Now,
$$\phi_i(x)=\phi_{i-1}\circ t_{\phi_{i-1}^{-1}(y_i)}(x)=\phi_{i-1}\circ \phi_{i-1}^{-1}(y_i)=y_i$$
and
$$\phi_i(y)=\phi_{i-1}\circ t_{\phi_{i-1}^{-1}(y_i)}(y)=\phi_{i-1}(x)=y_{i-1}$$
as we wanted.
We may conclude that $\phi_n$ is the desired automorphism.
