Ranks, Subdegrees and Suborbital graphs of the product action of Affine Groups

The action of affine groups on Galois field has been studied. For instance, [ 3 ] studied the action of 𝐴𝑓𝑓 (𝑞 ) on Galois field 𝐺𝐹 (𝑞) for 𝑞 a power of prime 𝑝 . In this paper, the rank and subdegree of the direct product of affine groups over Galois field acting on the cartesian product of Galois field is determined. The application of the definition of the product action is used to achieve this. The ranks and subdegrees are used in determination of suborbital graph, the non-trivial suborbital graphs that correspond to this action have been constructed using Sims procedure and were found to have a girth of 0, 3, 4 and 6.


Affine group
( ) over Galois field ( ) is a group of all transformations of the form + , where , ∈ ( ) and ≠ 0, these elements can be viewed as ( 0 1 ) . Let a group act transitively on a set Ω and the stabilizer of in for ∈ Ω, then ( ) is the Orbit of in . The -orbits are called suborbits of denoted by Δ and the number of them is the rank while the length of the suborbits is called the subdegrees of on Ω .
The concept of suborbital graphs of non-trivial suborbits of a permutation group was introduced by Sims. [2] Investigated some properties of the action of the stabilizer of ∞ in (modulargroup) acting on the set of integers is transitive and imprimitive. Moreover, if , ∈ with ≠ then the suborbital ( , ) and ( , ) are disjoint. The suborbital graph (0, ) has | | components and it is paired with ( , 0). [4] studied the action of (2, ) on the cosets of its cyclic subgroup ( −1) , where = ( − 1,2). The action was found to be imprimitive and the number of the self-paired suborbital is + 2, + 3 and + 1, for = 2, ≅ 1 4 and ≅ 1 4 respectively.

Preliminary Notes
Definitions 2.1 A group is said to be transitive on a set Ω if for each pair of points , ∈ Ω, there correspond ∈ such that = . In other words the action is transitive if it has only one orbit.

Definition 2.3
A graph ( , ) is said to be bipartite if can be partitioned into two subsets 1 and 2 such that the edges join the two vertices from different subsets and no edge joining vertices in the same subsets.

Theorem 2.5
Let be group acting on set Ω . Then, , for all ∈ Ω Theorem 2.6 [3] The suborbits of are of the form Corollary 2.7 [3] Let act on Ω . The subdegrees are of the form 1 and ( − 1) ; thus the rank of is two.

Theorem 2.9
A transitive action is said to be primitive if and only if each of the non-trivial suborbital graph is connected.

Theorem 2.10
The chromatic number of a bipartite graph is 2.

Theorem 2.11
Let be a connected undirected graph. Then is Eurelian if and only if every vertex is of even degree.
Theorem 2.12 [5] Let be a suborbital graph of a transitive action. Then all disconnected components of are isomorphic.

Proof.
Let This can also be proved by a method applied by [3].
Hence rank is 4 and subdegree are 1, on The suborbital corresponding to ∆ is given by, The suborbital graph corresponding to suborbital is formed by taking Ω 1 × Ω 2 × Ω 3 as the vertex set and directed edges from ( , , ) to ( , , ), if and only if (( , , ), ( , , )) ∈ . The suborbital graph corresponding to ∆ 0 is the null graph.
The construction of suborbital graphs corresponding to the action of
is the set of all the vertices in 3 , thus . Thus we get Equation (3.12).

Proof.
By Theorem [5], all the components in a suborbital graph are isomorphic. Applying Lemma 3.6, Since each of the graphs has 1 2 vertices, there are 1 components for 1 , 2 components for 2 and 1 component for 3 .

Lemma 3.8
For the action of 3 is bipartite.

Proof.
By Lemma 3.8, 3 is bipartite and using Theorem 2.10, chromatic number of a bipartite graph is 2.

Proof.
Let 3 be bipartite and be a primitive element in 4 ) → (0, 0), are all circuits of even length by Equation (3.8). Thus the girth is 4 and 6 respectively since 4 and 6 are the smallest length of a cycle respectively. Note that if 2 = 3, the results is as shown in Fig 3.6 Theorem 3.11 The suborbital graph 1 and 2 corresponding to the action of , has a girth of 0 when 1 = 2 = 2 and a girth of 3 when 1 = 2 > 2.

Proof.
From the Theorem 3.7, there are some non-trivial suborbital graphs which are disconnected. Thus from Theorem 2.9 the action is imprimitive.
. The suborbital 1 , 2 and 3 corresponding to the suborbit Δ 1 , Δ 2 and Δ 3 is, (3.14) Using Equations (3.13), (3.14) and (3.15), the suborbital graphs Γ 1 , Γ 2 and Γ 3 will be constructed as in Fig. 1, Fig.  2 and Fig. 3 respectively.   The suborbital graph 1 , 2 and 3 are all undirected, regular of degree1, has a girth of 0 and they are disconnected each with 2 components and each component has diameter 1 . The suborbital 1 , 2 and 3 corresponding to the suborbit Δ 1 , Δ 2 and Δ 3 is,  The suborbital graph 1 is undirected, regular of degree 2, has a girth of 3 and it is disconnected. It has The suborbital graph 1 is undirected, regular of degree 2, has a girth of 3 and it is disconnected. It has 2 connected components each with a diameter of 1. The suborbital graph 3 is undirected, connected, regular of degree 2 hence Eurelian, has a girth of 6 and a diameter of 2.