Projective characters of degree one and the inflation-restriction sequence

R. J. Higgs
1989 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
more » ... 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 [3] 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.
doi:10.1017/s1446788700030731 fatcat:xxjteegvn5cxbd7f23o7aamzge