Normality Of Tetravalent Cayley Graph Of One Class Of Groups Of Order P～4

Let G be a finite 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 to have vertex set V(Cay(G, S))= G and edge set E(Cay(G,S)) = {(g, sg) | g ∈ G, s ∈ S}. If S^-1 = S then Cay(G,S), called a Cayley graph, is viewed as an undirected graph by identifying two oppositely directed edges with one undirected edge. It is seen that a Cayley digraph Cay(G, S) is connected if and only if S generates G, and Aut(G, S) ={a ∈ Aut(G) | S^a = S} is a subgroup of the automorphism group Aut(Cay(G, S)) of Cay(G, S). Let g ∈ G. Define a map R(g) by x → xg, ?_x G. It is easy to show that R(g) is an automorphism of the Cayley digraph Cay(G, S) and the group R{G) = {R(g) | g ∈G}, called the right regular representation of G, is a subgroup of Aut(Cay(G, S)). A Cayley (di)graph Cay(G, S) is called normal if R{G) is normal in Aut(Cay(G, S)). Let p be an odd prime, G = (a, b|a^p3 = b^p = 1, a^b = a^1+p2). In this paper, we prove that all tetravalent connected Cayley graphs of G are normal by group theory and combinatorial theory, furthermore, we obtain two GRRs of G of valency 4.