Ranks, Subdegrees, Suborbital Graphs and Cycle Indices Associated with the Product Action of Affine Groups

Abstract

Many scholars have studied the ranks, subdegrees, cycle index and graphs of the action of the groups Cn,Dn and Aff(q) on a set X, where X = {1, 2, . . . , n} leaving out product actions. Recently, Kangogo (2015) studied the action of affine group over Galois field. The action of Aff(q1) × Aff(q2) on GF(q1) × GF(q2) and Aff(q1) × Aff(q2) × Aff(q3) on GF(q1) × GF(q2) × GF(q3) has not been studied. Using the definition of product action of orbits, the properties of the action of Aff(q1) × Aff(q2) on GF(q1)×GF(q2) and Aff(q1)×Aff(q2)×Aff(q3) on GF(q1)×GF(q2)× GF(q3) were studied and the rank was found to be 2k, where k = 2, 3 is the number of affine groups in the cross product. The subdegrees were found to be 1, (q1−1), (q2− 1), (q1 − 1)(q2 − 1) and 1, (q1 − 1), (q2 − 1), (q3 − 1), (q1 − 1)(q2 − 1), (q1 − 1)(q3 − 1), (q2−1)(q3−1), (q1−1)(q2−1)(q3−1) respectively. The corresponding non trivial graphs of Aff(q1)×Aff(q2) on GF(q1)×GF(q2) and Aff(q1)×Aff(q2)×Aff(q3) on GF(q1) × GF(q2) × GF(q3) were constructed using Sim’s procedure and were found to have a girth of 0, 3, 6 and 0, 3, 4, 6 respectively. Finally, cycle index were determined by first determining the cycle index of Aff(q) acting on GF(q) and then using multiplication of monomials to get the cycle index of the product action. The cycle indices have applications in chemistry when counting isomers.The graphs constructed provide useful information to graph theorist. Connectivity in graphs helps biologists to explain how the different parts of the brain are connected. The results have been represented in form of theorems and graphs.