Irreducible projective representations of finite groups

Any free presentation for a finite group G may be used to construct an infinite group F having G as quotient modulo a central subgroup, having finite commutator subgroup F' determined up to isomorphism by G, and having the projective lifting property for G over all fields. This thesis is concerned with the study of those irreducible representations of F which arise as lifts of irreducible projective representations of G over fields of characteristic zero.; If k is such a field, we obtain a bijective correspondence between the set of primitive central idempotents of the group algebra kF and the set of F-orbits of irreducible k-characters of F'. In the case where k is algebraically closed, this correspondence extends to the set of projective equivalence classes of irreducible projective k-representations of G.; In general the group algebra kF embeds in a completely reducible ring KF having dimension G&vbm0;H2 G,Cx&vbm0; over a purely transcendental field extension K of k. Analysis of the simple components of KF yields information on the general structure of certain simple k-algebras which appear as homomorphic images of kF, and on possible values of their Schur index and degree. These algebras determine irreducible projective representations of G over k, since they also appear as simple components of twisted group rings of G over k.; The problem of realizability of projective representations over small fields is considered in the light of the close connection between the equivalence classes of irreducible projective C -representations of G and the F-orbits of absolutely irreducible characters of F'. In particular it is shown that if the field k⊆C is an ordinary splitting field for the finite group F ', then every complex projective representation of G is projectively realizable in k.; Finally a detailed discussion of the irreducible projective representations of finite metacyclic groups over subfields of the field of complex numbers is included.