On projective representations on finite abelian group

Abstract

Saeed [11] has considered Schur multipliers of some of the finite abelian groups.The study of the schur multipliers of abelian groups is the first step in the studying of the projective representations of such groups. Our objective here is to determine the Inequivalent Irreducible projective representations of these groups which correspond to certain classes of factor sets. Let On' denote the direct product of n cyclic m groups C of order m. Then in [9] and [10] the a-regular classes have been determined; these being the classes at which non trivial projective representations with factor set a take on non zero character values. Here we review these results, and determine the Inequivalent Irreducible characters corresponding to these a-regular classes. In particular, a complete set of irreducible inequivalent projective characters is obtained for these classes. The following is a brief description of how the work in the sequel has been organised. Chapter one gives the basic facts about factor sets and projective representations of finite groups together with some of their properties. The concepts of schur multipliers and twisted group algebras are also considered. The central(vii)and stem extensions of finite groups are discussed in chapter two; while chapter three is concerned with projective character theory. Here the interest is in reviewing those properties of projective characters which are analogous to those of ordinary characters. Finally the work in the previous chapters is applied in chapter four to obtain the irreducible projective characters of certain finite abelian groups; and the results follow the works of Morris and Saeed (c.f [8], [9], [10] and [11].)