On the Character Table of the Multiplicative Group of a Finite Nearfield

This study investigates the representation theory of the multiplicative group of a finite nearfield, with a specific focus on the group <img src=image/13440431_01.gif>, arising from Dickson nearfields. Nearfields, generalizations of fields lacking one of the two distributive laws, pose interesting challenges in algebraic structures due to their noncommutativity and asymmetry. Their multiplicative groups, particularly those derived from Dickson nearfields, have a metacyclic structure that allows for intricate analysis using representation theory. The purpose of this research is to construct and analyze the character tables of these multiplicative groups. We explore foundational concepts such as Schur's lemma, Maschke's theorem, and orthogonality relations, and apply them within the context of finite nearfields. A combination of theoretical and computational methodologies is employed. Specifically, we make use of the Magma algebra system to build and verify character tables for <img src=image/13440431_01.gif> for small values of <img src=image/13440431_02.gif> and <img src=image/13440431_03.gif>, including detailed derivations for <img src=image/13440431_04.gif> and <img src=image/13440431_05.gif>. The principal results include explicit character tables, classifications of conjugacy classes, degrees of irreducible representations, and a generalization framework for any pair (<img src=image/13440431_06.gif>). The research demonstrates that the multiplicative groups <img src=image/13440431_01.gif> are not only metacyclic but also nilpotent and solvable under specific conditions. It further establishes structural conditions that determine the conjugacy class partitioning and representation degrees. Our conclusions highlight the deep connection between nearfield algebraic properties and finite group representation theory. This work contributes to the theoretical development of group representations in non-classical algebraic settings and opens doors to applications in geometry, coding theory, and cryptography. Practically, the use of Magma facilitates automated representation analysis, and socially, this research promotes a broader understanding of non-traditional algebraic systems. While the current study is computationally verified for small instances, future work will aim to optimize algorithms for handling larger <img src=image/13440431_01.gif> structures, enhancing scalability and applicability.