The normality of Cayley graphs of simple group \(A_5\) with small valencies (Q2721904)

Let \(G\) be a finite group and \(S\) a subset of \(G\) not containing the identity element 1. The Cayley digraph \(X=\text{Cay}(G,S)\) is the directed graph with vertex set \(V(X)=G\) and edge set \(E(X)=\{(g,sg): g\in G\), \(s\in S\}\). For each \(g\in G\), the mapping \(g_R: x\mapsto xg\) is an automorphism of \(X\), and the set \(G_R=\{g_R : g\in G\}\) is a subgroup of \(\text{ Aut}(X)\), the automorphism group of \(X\). The group \(G_R\) acts regularly on \(V(X)\), and it is called the right regular representation of \(G\). The Cayley digraph \(X\) is said to be normal if \(G_R\) is a normal subgroup of \(\Aut(X)\). In this paper it is shown that if \(G\) is the alternating group \(A_5\) and \(S\) is either a \(2\)-generating set of \(G\) or a \(3\)-generating set of \(G\), then \(X=\text{Cay}(G,S)\) is a normal Cayley digraph.