Automorphism groups of Cayley digraphs

Abstract

Let G be a group and S ⊂ G with 1 � 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 ∈ G , there is a directed edge from x to y whenever yx −1 ∈ S.I fS −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 ∈ 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(g)|g ∈ G} is a normal 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. Throughout this paper graphs or digraphs (directed graphs) are finite and simple unless specified otherwise. For a (di)graph X , we denote by V (X ), E(X ) and Aut(X ) the vertex set, the edge set and the automorphism group of X , respectively. A (di)graph is said to be vertex-transitive or edge- transitive if Aut(X ) acts transitively on V (X ) or E(X ), respectively. Note that for an (undirected) graph X , each edge {u, v} of X gives two ordered pairs (u, v) and (v, u), called arcs of X. Thus we sometimes, if necessary, view a graph X as a digraph. Let G be a group and S a subset of G such that 1 �∈ S. The Cayley digraph Cay(G, S) on G with respect to S is defined as the directed graph with vertex set G and edge set {(g, sg) | g ∈ G, s ∈ S}. For a Cayley digraph X = Cay(G, S), we always call | S | the valency of X for convenience. If S is symmetric, that is, if S −1 ={ s −1 | s ∈ S} is equal to S, then Cay(G, S) can be viewed as an undirected graph by identifying two oppositely directed edges with one undirected edge. We sometimes call a Cayley digraph Cay(G, S) a Cayley graph if S is symmetric, and say Cay(G, S) a directed Cayley graph to emphasize S −1 � S. Let X = Cay(G, S) be a Cayley digraph. Consider the action of G on V (X ) by right multiplica- tion. Then every element g ∈ G induces naturally an automorphism R(g) of X by mapping each vertex x to xg. Set R(G) ={ R( g) | g ∈ G}. Then R(G) is a subgroup of Aut(X ) and R(G) ∼ G. Thus X is a vertex-transitive digraph. Clearly, R(G) acts regularly on vertices, that is, R(G) is transitive on vertices and only the identity element of R(G) fixes any given vertex. Further, it is well-known that a digraph Y is isomorphic to a Cayley digraph on some group G if and only if its automorphism group contains a subgroup isomorphic to G, acting regularly on the vertices of Y (see (5, Lemma 16.3)). Noting that R(G) is regular on V (X ), it implies Aut(X ) = R(G)Aut(X )1.