Normal Edge-Transitive Cayley Graphs of the Group U 6 n

Let G be a group and let S be a subset of G. The Cayley graph of G with respect to S is a graph with the group G as the vertex set and (x, y) is an arc of the graph, if and only if for some s ∈ S we have y = sx. We denote the Cayley graph by Γ = Cay(G, S). SinceG is a group, the Cayley graph cannot have any parallel edges. In addition we assume that the subset Sdoes not contain the identity element of the groupG to avoid having loops in the Cayley graph and also to be an inverse closed set; that is, S = S, to have an undirected Cayley graph. Therefore, we focus on simple Cayley graphs. Let Γ = (V, E) be a simple graph with vertex set V and edge set E. In this paper, we denote the edge, joining the vertices x and y by {x, y}. An automorphism of the graph Γ is a permutation on the vertex set of Γ which preserves the edges. The set of all automorphisms of Γ forms a group with the composition of maps as the binary operation and is denoted by Aut(Γ). Since Aut(Γ) is a permutation group of V, it acts on V as well as E in the usual way. Γ is called vertex-transitive, edge-transitive, or arc-transitive, iff Aut(Γ) acts transitively on the set of vertices, edges, or arcs of Γ, respectively. Γ is called half-arc-transitive iff it is vertexand edge-transitive, but not arc-transitive. LetG be a finite group and let S be an inverse closed subset of G that does not contain the identity. Set Γ = Cay(G, S). Γ is connected iff S is a generating set of the group G. For g ∈ G, define the mapping ρ g : G → G by ρ g (x) = xg, for all x ∈ G.


Introduction
Let  be a group and let  be a subset of .The Cayley graph of  with respect to  is a graph with the group  as the vertex set and (, ) is an arc of the graph, if and only if for some  ∈  we have  = .We denote the Cayley graph by Γ = Cay(, ).Since  is a group, the Cayley graph cannot have any parallel edges.In addition we assume that the subset  does not contain the identity element of the group  to avoid having loops in the Cayley graph and also to be an inverse closed set; that is,  =  −1 , to have an undirected Cayley graph.Therefore, we focus on simple Cayley graphs.
Let Γ = (, ) be a simple graph with vertex set  and edge set .In this paper, we denote the edge, joining the vertices  and  by {, }.An automorphism of the graph Γ is a permutation on the vertex set of Γ which preserves the edges.The set of all automorphisms of Γ forms a group with the composition of maps as the binary operation and is denoted by Aut(Γ).Since Aut(Γ) is a permutation group of , it acts on  as well as  in the usual way.Γ is called vertex-transitive, edge-transitive, or arc-transitive, iff Aut(Γ) acts transitively on the set of vertices, edges, or arcs of Γ, respectively.Γ is called half-arc-transitive iff it is vertex-and edge-transitive, but not arc-transitive.
Let  be a finite group and let  be an inverse closed subset of  that does not contain the identity.Set Γ = Cay(, ).Γ is connected iff  is a generating set of the group .For  ∈ , define the mapping   :  →  by   () = , for all  ∈ .  ∈ Aut(Γ) for every  ∈ , and hence () = {  |  ∈ } is a regular subgroup of Aut(Γ) isomorphic to , forcing Γ to be a vertex-transitive graph.
In [2], the graph Γ is called normal if () is a normal subgroup of Aut(Γ).And after that the normality of Cayley graphs has been extensively studied from different points of view.Among them, finding and classifying the normal Cayley graphs were an essential problem, since in normal Cayley graphs we know the exact automorphism group of the graph.
It was conjectured in [2] that most Cayley graphs are normal.For example, in [3] the authors determined all possible nonnormal (and, as a consequence, normal) Cayley graphs of groups of order 2 2 , and lots of other authors have done similar works for groups of orders , 2, 4,  2 , and  [4][5][6][7][8].
Another concept which was similar to the above one is introduced by Praeger in [9] in which a Cayley graph Γ = Cay(, ) of a group  with respect to  is called normal edge-transitive or arc-transitive if  Aut(Γ) (()) acts on the edges or arcs of Γ, respectively, and it is called normal halfarc-transitive Cayley graph if it is normal edge-transitive Cayley graph which is not normal arc-transitive.Obviously, any normal edge-transitive Cayley graph is edge-transitive.

International Journal of Combinatorics
Thus, this concept talks about the symmetric properties of a Cayley graph.
The latter concepts also were considered very much in the literature.For example, Alaeiyan in [10] found a class of normal edge-transitive Cayley graphs of abelian groups or, in [11], authors found all normal edge-transitive Cayley graphs of order 4 and as a consequence found a class of normal half-arc-transitive Cayley graphs which rarely happens.This motivated us to consider the Cayley graphs of groups  6 and classify all normal edge-transitive Cayley graphs of groups  6 .

Preliminary Results
One of the principle theorems that helps us to connect the group properties of a group  and normal edge-transitivity of Γ = Cay(, ) is the following theorem which is proved in [9].Theorem 1.Let Γ = (, ) be a connected Cayley graph on .Then, Γ is normal edge-transitive if and only if (, ) is either transitive on  or has two orbits in  in the form of  and  −1 , where  is a nonempty subset of  such that  =  ∪  −1 .
The following corollary comes from Theorem 1 which is also mentioned in [11].
Corollary 2. Let Γ = (, ) and let  be the subset of all involutions of the group .If  does not generate the group  and Γ is connected normal edge-transitive, then the valency of Γ is even.
Theorem 1 with a result in [12] characterizes arc-transitive and half-arc-transitive Cayley graphs as described in the following.
Theorem 3. Let Γ = (, ) be a connected normal edgetransitive Cayley graph.Γ is normal arc-transitive if (, ) acts transitively on  and Γ is normal half-arc-transitive if  =  ∪  −1 , where  is an orbit of the action of (, ) on .

Group 𝑈 6𝑛
Group  6 has the presentation which we can write its elements as follows: One can see that the group  6 has order 6.
In Theorem 1, there are some relations between the normal edge-transitive Cayley graphs and the automorphism group of the relying group.Therefore, first we will find the automorphism group of the group  6 .Theorem 4. Every automorphism of the group  6 is in the form of  , :  6 →  6 which sends  to   and  to   , or  ,, :  6 →  6 sends  to     and  to   , where (, 2) = 1 for 0 ⩽  ⩽ 2 − 1 and 1 ⩽ ,  ⩽ 2.
Proof.First of all we find some relations between elements of the group  6 .
if the order of  =    is , for odd , then  = 2 for some integer .Since order of  is 2, we should have But  should be the least integer satisfying the condition and implies  = /(, ); that is, where (⋅, ⋅) is the greatest common divisor.Similar argument can apply for    2 in the case  is odd to obtain Now for even , assume  = 2 for some integer  and (  ) =  = 3 for some integer .Thus, we have Compare it with the order of ; we get Since 3 | 3, one can conclude that [/(, ), 3] | 3, where [⋅, ⋅] is the least common multiple, and finally we obtain Similar argument can be discussed to prove Suppose  is an automorphism of the group  6 ; thus,  preserves the order of elements; hence, (()) = () = 2 and (()) = () = 3.
By the order of elements of  6 , if we define the following sets (observe that in this case  is odd) by the order of elements of the group  6 , we can say that () ∈ ∪ and () ∈ ∪∪.Let  = () and  = ().But all of the following cases which may happen for  and  yield a contradiction.
Therefore, the only cases that may happen are the cases  ∈ ,  ∈  and  ∈ ,  ∈  and the theorem is proved.
If  is even, then If  is even or odd, then  , (  ) =   and  ,, (  ) =   .Therefore, the orbit of   for an even  is {  |  is even, (, 2) = 1} and for odd  is {    |  is odd, 0 ⩽  ⩽ 2}.The orbit of     for an even  is {    |  is even, 1 ⩽  ⩽ 2} and for odd  is {    |  is odd, 0 ⩽  ⩽ 2}.And finally the orbit of  is {,  2 }.Now we bring a sufficient condition under which a Cayley graph of group  6 can be normal edge-transitive.Proof.From Corollary 2, || is even.