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

Siahi Maxwell Agwanda,Ireri Kamuti,Patrick Kimani

doi:10.24297/jam.v19i.8891

Siahi Maxwell Agwanda, Ireri Kamuti + Show 1 more

Open Access

https://doi.org/10.24297/jam.v19i.8891

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Abstract

The action of affine groups on Galois field has been studied. For instance, studied the action of on Galois field for a power of prime. 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.

Highlights

Affine group Aff (q ) over Galois field GF (q) is a group of all transformations of the form ax + b, where a, b ∈ GF(q) and a ≠ 0, these elements can be viewed as (a0 b1)
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 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

Summary

Introduction

Affine group Aff (q ) over Galois field GF (q) is a group of all transformations of the form ax + b, where a, b ∈ GF(q) and a ≠ 0, these elements can be viewed as (a0 b1). Let a group G act transitively on a set Ω and Ga the stabilizer of a in G for a ∈ Ω, OrbG(a) is the Orbit of a in G. The Ga-orbits are called suborbits of G denoted by Δi and the number of them is the rank while the length of the suborbits is called the subdegrees of G on Ω. [2] Investigated some properties of the action of the stabilizer of ∞ in Γ(modulargroup) acting on the set of integers is transitive and imprimitive. The suborbital graph Γ(0, x) has |x| components and it is paired with Γ(x, 0). The action was found to be k imprimitive and the number of the self-paired suborbital is q + 2, q + 3 and q + 1, for p = 2, q ≅ 1mod and q ≅ 1mod respectively

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