I am reading "Algebraic Graph Theory" by Biggs 1974. In the section about symmetric graphs, it is stated that:
A vertex transitive graph $X$ is symmetric, if and only if each vertex-stabilizer $G_v$ is transitive on the set of vertices adjacent to $v$.
I see why this is a necessary condition, but I do not see why this is a sufficient condition. Can anybody give a short proof of this Lemma?