The group algebra of a finite group: a structural study

Abstract

The present work is a study of the structure of the group of an arbitrary finite group over a given field of characteristic zero. A classical result due to Maschkehas it that the group algebra is semi simple. This work nevertheless starts with the result that the group algebra does have a direct sum decomposition into simple rings, the point being that the proofs do not employ the usual approach of using the minimum condition. The simple components are then studied by considering them as elements of the group of field. Chapter one and two contain respectively the direct sum decomposition of the group algebra and a characterization of the summands. In chapter three the Brauer group is introduced and subsequently the Schur subgroup (of the Brauer group) whose elements arises as simple components of a group of algebra. Also introduced in chapter three is the subgrpoup of the Brauer group whose elements arises as cyclotomic algebras and whose importance stems from the fundamentals results (Brauer-Witt) that it (the subgroup) exactly equals the Schur subgroup. Finally chapter four focuses on the Schur subgroup of an algebraic number field. At the end of the work appear four appendices which are really an extension of the three pages on Notations and definitions. Full accounts of the work set out there can be obtained from references (2) and (4)