Generic central extensions and projective representations of finite groups

Abstract

Any free presentation for the finite group G G determines a central extension ( R , F ) (R,F) for G G having the projective lifting property for G G over any field k k . The irreducible representations of F F which arise as lifts of irreducible projective representations of G G are investigated by considering the structure of the group algebra k F kF , which is greatly influenced by the fact that the set of torsion elements of F F is equal to its commutator subgroup and, in particular, is finite. A correspondence between projective equivalence classes of absolutely irreducible projective representations of G G and F F -orbits of absolutely irreducible characters of F ′ F’ is established and employed in a discussion of realizability of projective representations over small fields.