Normality of 2-Cayley digraphs

A digraph Gamma is called a 2-Cayley digraph over a group G, if there exists a semiregular subgroup R-G of Aut(Gamma) isomorphic to G with two orbits. We say that Gamma is normal if RG is a normal subgroup of Aut(Gamma). In this paper, we determine the normalizer of RG in Aut(Gamma). 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 not equal Q(8) x 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. (C) 2014 Elsevier B.V. All rights reserved.