Representation Groups for the Symmetric Group

Abstract In this chapter, we shall determine the Schur multiplier of the symmetric group Sn, and give presentations for it representation groups. To find an upper bound for M(Sn), we shall present a portion of Schur’s method to calculate M(G) starting from a presentation of G. For this, two basic results in group theory, Theorems 2.1 and 2.3 below, are quoted without proof. Alternatively, there is a direct approach to this upper bound which uses the twisted group algebra and the existence of a projective representation for any given cocycle; for a treatment, see Jozefiak (1989), pp. 198-200. The first quoted result consists of some consequences of the classification theorem for finitely generated abelian groups.