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.