Generic Central Extensions and Projective Representations of Finite Groups

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