Let G be a finite group generated by a and b. Let s1, s2, . . . , sn be the arcs of a Ha...

Let G be a finite group generated by a and b. Let s₁, s₂, . . . , s_n be the arcs of a Hamiltonian circuit in the digraph Cay({a, b}:G). We say that the vertex s₁s₂ ….s_i travels by a if s_i+1 = a. Show that if a vertex x travels by a, then every vertex in the travels by a.