On Cyclotomic Cosets as Orbit Structures under the Action of Cyclic Subgroups of GL(d,q)
Pancras Onyango Ongili *
Department of Pure and Applied Sciences, Kirinyaga University, P. O. Box 143-10300, Kerugoya, Kenya and Department of Mathematics, Kiriri Women’s University of Science And Technology, P. O. Box 49274 – 00100, Nairobi, Kenya.
*Author to whom correspondence should be addressed.
Abstract
Cyclotomic cosets are classical algebraic objects that arise naturally in modular arithmetic, finite field theory, and coding theory, especially in the study of polynomial factorization and cyclic codes. Traditionally, they are defined arithmetically through the formation of sets of the form Ci = {i · pj mod n | j ≥ 0}, and their study has largely remained within that framework. However, this approach does not fully capture the group-action structure inherent in their formation. In particular, the interpretation of cyclotomic cosets as orbits under subgroup actions has not been formally developed in a way that extends to general linear groups over finite fields. This creates a theoretical gap between classical cyclotomy and the broader framework of orbit theory and linear group actions. This paper uses methods of mathematical proofs to addresses this gap by reinterpreting cyclotomic cosets as orbit structures. This idea is further extended to the natural action of subgroups of GL(d,q) on the vector space \(\mathbb{F}_q^d\) , thereby defining a generalized cyclotomic orbit structure in higher dimensions. It is established that
Orb⟨p⟩(i) = {pj · i | j ≥ 0} = {i · pj mod n | j ≥ 0}, ⇒ Ci = Orb⟨p⟩(i),
showing that classical cyclotomic cosets are precisely orbit structures arising from cyclic subgroup actions. The findings further reveal that for the action of a subgroup H ≤ GL(d,q) on \(\mathbb{F}_q^d\), the cyclotomic orbit CH(v) = {h · v | h ∈ H} satisfies |CH(v)| = [H : StabH(v)], showing that the size of each cyclotomic orbit is determined by the index of its stabilizer. This establishes a direct connection between cyclotomic structures and the Orbit–Stabilizer Theorem.This formulation places classical cyclotomic cosets within a wider orbit theoretic setting and provides a new perspective for studying cyclotomic behavior through linear actions, contributing to the further study of orbit structures, invariants, and their advanced applications in finite fields, and group theory.
Keywords: Cyclotomic orbit structures, cyclotomic cosets, group actions, orbits, general linear groups, GL(d,q), finite fields, Orbit–Stabilizer Theorem