Group Representations, Nilpotent Algebras and Finite Algebra Groups

Abstract

The purpose of this thesis is introduce the reader to representations of finite algebra groups and summarise some key results concerning such representations. The work that follows begins by reviewing key properties of finite group representations and nilpotent algebras. The representations studied are for the most part complex representations, however much of the theory applies equally to many other Fields. We follow this discussion with an introduction to finite algebra groups by exploring the relationship that exists between these groups and the Jacobson Radical of algebras over finite fields. Then we consider the work of Zoltan Halasi in significant detail and attempt to clarify some of the ambiguites a reader may face in reading his paper. In addition, we prove and state many of the excluded facts and results on which his arguments rely. Finally, we conclude by analysing the irreducible representations of a specific class of finite algebra groups. This is our working example. In doing so, we highlight how the work of Halasi simplifies the search for irreducible representations. We end the final section by introducing a non-finite algebra group that shares many similarities to the class of groups considered in our working example. Determining whether or not an analagous theorem to the one proved by Halasi holds for these groups is an open problem. It is the belief of the author that a person who is interested in exploring the irreducible representations of such groups may find this thesis a solid introduction. Show more