Projective characters of degree one and the inflation-restriction sequence
Journal of the Australian Mathematical Society
Let G be a finite group, a be a fixed cocycle of G and Proj(G, a) denote the set of irreducible projective characters of G lying over the cocycle a. Suppose N is a normal subgroup of G. Then the author shows that there exists a Ginvariant element of Proj(JV, apt) of degree 1 if and only if [a] is an element of the image of the inflation homomorphism from M(G/N) into M(G), where M(G) denotes the Schur multiplier of G. However in many situations one can produce such G-invariant characters where
... characters where it is not intrinsically obvious that the cocycle could be inflated. Because of this the author obtains a restatement of his original result using the Lyndon-Hochschild-Serre exact sequence of cohomology. This restatement not only resolves the apparent anomalies, but also yields as a corollary the well-known fact that the inflation-restriction sequence is exact when N is perfect. 1980 Mathematics subject classification (Amer. Math. Soc): 20 C 25. Keywords and phrases: projective representations of finite groups. All groups, G, considered in this paper are finite and all representations of G are defined over the complex numbers. The reader unfamiliar with projective representations is referred to  for basic definitions and elementary results. The purpose of this paper is to investigate under which circumstances the following well-known corollary to Clifford's theorem can be generalized to projective characters.