Normal Cayley graphs of finite groups

Abstract

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≅ℤ4×ℤ2 orG≅Q 8×ℤ 2 (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.