A Symmetry of a graph X is a permutation of the vertices that also permutes the edges. The distances between vertices are preserved. Show that the symmetries of a graph are a permutation group,
So that means I need to show that the symmetries of a graph have the following properties:
Closure: if a,b are in the set of S then a.b is in the set of S
Associativity: if a,b,c are in the set S then a.(b.c)=(a.b).c
Identity: there exists i in the set S such that i.a=a.i=a for all a in S
Inverse for all a in S there exists an inverse b in S such that a.b=b.a=i
How do I prove any of these properties for the symmetries of a graph? I am really confused. We haven't seen any examples of how to prove these so far.