Automorphism groups of Cayley digraphs

Let G be a group and S subset of G with 1 is not an element of S. A Cayley digraph Cay(G, S) on G with respect to S is the digraph with vertex set G such that, for x,y is an element of G, there is a directed edge from x to y whenever yx(-1) is an element of S. If S-1 = S, then Cay(G, S) can be viewed as an (undirected) graph by identifying two directed edges (x,y) and (y,x) with one edge {x,y}. Let X = Cay(G, S) be a Cayley digraph. Then every element g is an element of G induces naturally an automorphism R(g) of X by mapping each vertex x to xg. The Cayley digraph Cay(G, S) is said to be normal if R(G) = {R(gy)vertical bar g is an element of G} is a non-nal subgroup of the automorphism group of X. In this paper we shall give a brief survey of recent results on automorphism groups of Cayley digraphs concentrating on the normality of Cayley digraphs.