AUTOMORPHISM GROUPS OF 4-VALENT CONNECTED CAYLEY GRAPHS OF p-GROUPS

Let G be a p-group (p odd prime) and let X=Cay(G,S) be a 4-valent connected Cayley graph. It is shown that if G has nilpotent class 2, then the automorphism group Aut(X) of X is isomorphic to the semidirect product GRAut(G, S), where GR is the right regular representation of G and Aut(G, S) is the subgroup of the automorphism group Aut(G) of G which fixes S setwise. However the result is not true if G has nilpotent class 3 and this paper provides a couterexample.