Normal Arc-transitive Cayley Graphs Of Order6p~2and Prime Valency

In order to better define the relationship generators and exposition group, A.Cayl-ey concept map proposed in1878and to study its properties. A graphΓ is called a Cayl-ey graph of a group G, if there is a subset S(?)G\{1}, with S=S-1:={g-1|g∈S}, such that VΓ=G, and two vertices g and h are a adjacent if and only if hg-1∈S. This Cayley graphΓ is denoted byΓ=Cay(G, S). If there is X≤Aut(Γ) such that X is transitive on t-he arc set of Γ, then Γ is called an X-arc transitive graph; if further the subgroup G is n-ormal inΓ, thenΓ is called X-normal arc transitive Cayley graph. In particular, if G is normal in Aut(Γ), then graph Γ is called a normal Cayley graph. The main purpose of t-his thesis is to characterize normal arc transitive Cayley graphs Γ of prime valency and order6p2and determine the structure of the corresponding underlying group, where p≥5is a prime. By analyzing normal quotient graph of Γ, we first prove that val(Γ)=3or5. SoΓ is either a Zp2-regular covering or Zp2-regular covering of the complete graph K6, or, a Zp2-regular covering or Zp2-regular covering of the complete bipartite graph K3,3. Further it is proved that no such graph Γ, if val(Γ)=5; and if val(Γ)=3, either Γ(?)G(k,p2) is a Zp2-regular covering of K3,3, where2≤k≤p2such that p2|k2+k+1, orΓ(?) EBp2is a Zp2-regular covering of K3,3. Moreover, the structure of underlying group are specifically determined.