Normal Cayley graphs of finite groups

LetG be a finite group and let S be a nonempty subset of G not containing the identity element 1. The Cayley (di) graph X = Cay(G, S) of G with respect to S is defined byV (X)=G, E (X)={(g,sg)|g∈G, s∈S} A Cayley (di) graph X = Cay (G,S) is said to be normal ifR(G) ?A = Aut (X). A group G is said to have a normal Cayley (di) graph if G has a subset S such that the Cayley (di) graph X = Cay (G, S) is normal. It is proved that every finite group G has a normal Cayley graph unlessG??4×?2 orG?Q 8×? 2 r (r?0) and that every finite group has a normal Cayley digraph, where Zm is the cyclic group of orderm and Q8 is the quaternion group of order 8.

Let G be a finite group and let S be a nonempty subset of G not containing the identity element 1. The Cayley (di)graph X=Cay(G,S) of G with respect to S is defined by V(X)=G,E(X)={(g,sg)g∈G,s∈S}. A Cayley (di)graph X=Cay(G,S) is said to be normal if R(G)A=Aut(X). A group G is said to have a normal Cayley (di)graph if G has a subset S such that the Cayley (di)graph X=Cay(G,S) is normal. It is proved that every finite group G has a normal Cayley graph unless GZ 4×Z 2 or GQ 8×Z r 2(r≥0) and that every finite group has a normal Cayley digraph, where Z m is the cyclic group of order m and Q 8 is the quaternion group of order 8.