Normality of 2-Cayley digraphs

A digraph ? is called a 2-Cayley digraph over a group G , if there exists a semiregular subgroup R G of Aut ( ? ) isomorphic to G with two orbits. We say that ? is normal if R G is a normal subgroup of Aut ( ? ) . In this paper, we determine the normalizer of R G in Aut ( ? ) . We show that the automorphism group of each normal 2-Cayley digraph over a group with solvable automorphism group, is solvable. We prove that for each finite group G ? Q 8 i? Z 2 r , r ? 0 , where Q 8 is the quaternion group of order 8 and Z 2 is the cyclic group of order 2, there exists a normal 2-Cayley graph over G and that every finite group has a normal 2-Cayley digraph.