J Austral Math Soc Ser A 46 pp272--280, 1989.

Projective Characters of Degree One and the Inflation-Restriction Sequence

R. J. Higgs

Abstract

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 G-invariant element on Proj(N, a_N) 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 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

{1} ® M(G/N) ® M(G) ® M(N)

is exact when N is perfect.

1980 AMS Subject Classification (1985 Revision): 20C25

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Authors

R. J. Higgs

Department of Mathematics, University College Dublin, Belfield, Dublin 4, Ireland.